202110723-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G10-FY_Optimized

PRACTICE SHEET - 5 (PS-5) II. Short answer questions. 1. Find the value of ‘k’ for which the equation 9f² – 6kf + 4 = 0 has equal roots. 2. Find the roots of the equation x² + 4b² – 4bx – d² = 0. 3. Given that α and β are the roots of 1 – 7x + 5x² = 0, find the sum of their reciprocals. III. Long answer questions. 1. Three consecutive whole numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 192. Find the numbers. 2. The difference of two natural numbers is 25 and the difference of their reciprocals is 1 . Find the numbers. 24 37

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1) Raju is 4 years younger than Ravi. Five years from now the product of their ages will be equal to 8 times Raju’s current age. Express the problem mathematically. (2 Marks) 2) Determine if the given equations are quadratic equation. x3 + 4 x2 − 8 x +12 = x45 3 i) 5   ii) ( x + 5)( x − 3)(2x + 3) = x3 + 4 x (2 Marks) 3) Find the roots of the quadratic equation using factorisation method: (3 Marks) ( ) (2x − 3)2 −11x + 5 − x2 =0 4) Two consecutive whole numbers are to be selected such that the sum of the smaller number and the square of the larger number must be 55. Determine the two numbers using the method of completing the square.    (4 marks) 5) eFqinudatthioendiifstchreimy ianraenrteoafl.t he eq uatio n 2x2 − 43 x − 10 = x 2 + 43 x − 20 and hence find the roots of the 3 (4 Marks) 38

5. Arithmetic Progressions • Calculate the sum of the terms on an AP. Learning Outcome • Determine the number of terms in an AP. By the end of this chapter, a student will be able to: • Determine if a given series of numbers is an arithmetic progression. • Determine the value of any term of an AP. Concept Map ������������ Key Points This is called the general form of an AP. • If the AP has a finite number of terms then it is • Progression is a pattern of numbers following a definite rule. called as finite arithmetic progression and these Examples: 1, 4, 9, 16, ….. is progression of squares arithmetic progressions have a last term. If the AP of numbers has no last term, then it called as infinite arithmetic 2, 7, 12, 17,… is progression of numbers progressions. obtained by adding 5. • nth Term of an AP • Arithmetic Progression (AP) Let a1 , a2 , a3 , a3 ......,, an−1 , �an ,......be the terms of Arithmetic Progression is a list of numbers in which an AP and a be the first term and d be the common each term is obtained by adding a fixed number to difference of the AP. the preceding term except the first term. We have The fixed term which is added is called the common a1 = a difference of the AP. The common difference can be positive, negative or zero. a2 = a + d = a + (2 −1) d Example: a3 = a + 2d = a + (3 −1) d 2,5,8,11,14,…. First term is 2, common difference is 3  1,–2,–5,–8,–11,…First term is 1, common difference is –3 an = a + (n −1) d 3,3,3,3,3,3…. First term is 3, common difference is 0 The nth term of an AP with a as the first term and Let us denote the first term of an AP by a1 , second d as the common difference is given by term by a2 , third term by a3 , …., nth term by an . Let the common difference of the AP be d. The AP an = a + (n −1) d ba1e,cao2m, ea3s ,⊃ , an−2 , an−1 , �an So, a2 − a1 = a3 − a2 = … = an−1 − an−2 = an − an−1 = d • Sum of first n terms on an AP a, a + d, a + 2d, a + 3d,....... represents an AP where If a, d are the first term, common difference of an a is the first term and d is the common difference. arithmetic progression, then the sum of first n terms is given by S, S = n 2a + (n −1) d  2 If an = l is the last term of the AP, then S = n a + a + (n −1) d  2 ⇒ S = n (a +l) 2 39

5. Arithmetic Progressions Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Recognizing patterns in a series of PS – 1 numbers PS – 2 • Understanding Arithmetic Progressions PS – 3 PS – 4 Arithmetic Progressions • Finding the nth term of an AP PS - 5 Self Evaluation • Sum of first n terms of an AP Sheet Worksheet for “Arithmetic Progression” Evaluation with Self ---- Check or Peer Check* 40

PRACTICE SHEET - 1 (PS-1) 1. Find the missing numbers (x) from the list of numbers given below: i)  2, 4, 6, 8, x, 12, 14, 16….. ii)  3, 6, 9, 12, x, 18, 21, 24…… iii)  1, 2, 3, x, 8, 13, 21, 34, ….etc. iv)  1, 4, 9, 16, x, 36, 49, 81,….. 2. Find the relation between the lists of numbers given below: i)  1, 4, 7, 10, 13, 16, 19….. ii)  2, 5, 10, 17, 26, 37, 50, 65, 82, 101, …. iii)  1, 3, 9, 27, 81, 243 ............... iv)  3, 6, 12, 24, 48, 96, ........ PRACTICE SHEET - 2 (PS-2) 1. Write the first five terms of the AP using first term and common difference. i) a = 5 and d = 3 ii) a = −10 and d = −3 iii) a = 1 and d = 1 24 2. Check if the given numbers form an AP and determine the first term and the common difference. i) 1 , 2 , 3 , 2 2 , ……. 22 ii)  0, −1, −3, −5, −7, ….. 3. The heights of the plant are measured at the end of each month and the heights are 1.5 m, 2.5 m, 4.5 m and 6 m respectively. Determine the height of the plant at the end of fifth month. 4. Determine the value of p if the given series of numbers form an A.P. i)  0.1, 0.65, p, 1.75, ……  ii) p, 12 , 27 , 48 ,…. 5. Five numbers are picked from an AP and are given below: …., 3, 10, 17, 24, 31, …. Find 2 numbers before 2 and 2 numbers after 31 of the A.P. 6. If a1 , a2 , a3 and a4 are the four consecutive terms of an A.P. Prove that. i) a1 + a4 = a2 + a3   ii) a2 = a1 + a3 2 iii)  a3 = a2 + a3 + a4 3 41

PRACTICE SHEET - 3 (PS-3) 1. Find the 10th and 20th terms of the AP if their first term and common difference are as follows: i) a = 20 and d = 1 ii)  a = 1 and d = 20 iii) a = 20 and d = −1 iv)  a = −20 and d = 1 2. If 3 and 17 are 3rd and 6th terms of an AP, find the first term, common difference and also the terms between them. 3. A young man joined as a clerk in a bank for a salary of Rs. 50 per month. Every year his monthly salary was increased by Rs. 60. What would be his monthly salary after 25 years? 4. Three consecutive terms of an AP are 7, 11, 15. Find if 103 is part of the same AP and its position from 7. 5. Sudhir and Sunil joined two different companies. Sudhir got a starting salary of Rs. 500 per month and in every year his salary was increased by Rs. 100 per month. Sunil got a starting salary of Rs. 800 per month and each year his monthly salary was increased by Rs. 50. After 40 years the two friends retire. i) Check when would their salaries be equal and what is the salary when it becomes equal? ii) What was their salary at retirement and who was getting more salary? 6. If 7th term of an AP is 12 more than 4th term, how much will be 15th term more than 7th term and 10th term. 7. 150, 143, 136, ….. form an AP. Which term of the AP would be the first negative term? What is the 10th negative term of the AP? PRACTICE SHEET - 4 (PS-4) 1. Find the sum of the first ten terms of the AP given below: i)  5, 8, 11, ……  ii)  -3, –2, –1, 0, 1, ….. iii)  15, 22, 29, ….  iv)  -5, –10, –15, –20, …… 2. Raghu was saving money for his daughter’s medical education. By the tenth year from now he wanted to save a total of Rs. 11 lakhs. Currently his monthly salary is Rs. 15000 and he could save 1 of his salary. His total savings currently is Rs. 50000 only. Can Raghu save the required 3 money with the current salary for the next 10 years? What should be the increase in his salary each year so that he could save for his daughter’s education? 3. If the sum of first 50 numbers of an AP is zero then find the relation between the first term and common difference. Find the values of a if the common differences is i) 5  ii) 10  iii) -7  iv) 1 4. If the sum of 20 terms of an AP is 380 more than the first term and 10th term of the AP is 15 then find the first three terms on the AP. 5. Find the 10th term of an AP, if its 3rd term is 5 and 20th term is 8. 6. Find the sum of the following numbers: 155, 160, 165, ……., 290, 295. 42

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. Which of the following is the common difference of an A.P? (A) The difference of a term and its previous term. (B) The difference of a term and its next term. (C) The sum of a term and its previous term. (D) The ratio of a term and its previous term. 2. The nᵗʰ term of an A.P. with ‘a’ as its first term and ‘d’ as its common difference is given by _________. W(Ah) ticnh= a + (2n – 1)d s(aBm) ten = a + (n + 1)d give(nC)At.nP-.?1 = a + (n + 1)d (D) tn = a + (n – 1)d 3. among these is the throughout a (A) First term (B) Common difference (C) nth term (D) All of these 4. The sum of three terms of an A.P. with first term ‘a’ and common difference ‘d’ is given by ___________. n (B) S3 = 3 [a + d] (C) S3 = 3[2a + 2d] (D) S3 = 3[a + d] (A) Sⁿ = 2 [2a + (n – 1)d] 2 5. The sum to ‘n’ terms of an A.P. with first term ‘a’ and last term ‘  ’ is given by ___________. n [a +  ] n (C) Sn = n[2a +  ] (D) Sn = n[a +  ] (A) Sn = 2 (B) Sn = 2 [a + d] 6. The common difference of an A.P. can be ________________. (A) negative (B) positive (C) zero (D) All of these 7. Which of the following A.P.s have a negative common difference? (A) 145, 140, 135, 130,… (B) -73, - 70, - 67, - 64, … (C) -54, - 58, - 62, -66, … (D) Both (A) and (B) 8. The next term in the A.P. 12, 14, 16, … is ______. (A) 18 (B) 14 (C) 20 (D) 24 9. The 5th term of the A.P. 1, -1, -3, … is __________. (A) 7 (B) 4 (C) -7 (D) 5 10. The numbers that fill the blanks in order, in the A.P. 7, 13, ____ ,____, … are __________. (A) 19, 25 (B) 25, 19 (C) 17, 19 (D) 19, 17 II. Short answer questions. 1. Find an A.P. whose 4th term is 4 and the 8th term is 8. 2. How many 2 – digit numbers are divisible by 2? 3. The first term of an A.P. is 14 and the sum of its first 15 terms is 1365. Find the 25th term of the A.P. III. Long answer questions. 1. Find the sum of the first 20 terms of the A.P. whose nth term is given by 4 – 5n. 2. A factory produces 500 toy cars on the second day and 700 toy cars on the sixth day. The production in- creases uniformly by a fixed number every day. Find the number of toy cars produced on the first day and the total number of toy cars produced in first 6 days. 43

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins (2 Marks) 1. Find the next three terms if the given numbers form an AP: –10, –3, 4, 11, 18…. 2. Is it possible to find the last term of an AP given its first term and common difference? Explain. (1 Mark) 3. If 10th, 30th terms of an AP are 50, 80 respectively. Find the 50th term of the AP. (2 Marks) 4. Find the sum of the following numbers: (4 Marks) i)  First 50 odd numbers ii)  First 50 even numbers. iii)  990, 979, 968, …., 22, 11. 5. Three alternate terms on an AP are 5, 15, 25. If the first term of the AP is –50 then, (4 Marks) i)  Find the sum of the first 10 terms. ii)  Sum of first 30 terms. iii)  Sum of 11th to 20th terms. 6. Two arithmetic progressions have the same common difference between the terms. If their first terms are 8, 5 then find the difference of 20th terms of the two AP. (2 Marks) 44

6. Triangles Learning Outcome By the end of this chapter, a student will be able to: • Determine the relation between lengths and • Determine the similarity between the given angles using principles of similar triangles. geometrical shapes. • Determine the areas of similar triangles. • Determine the relation between sides of figures • Apply Pythagoras theorem to find relations using basic proportional theorem. between sides and angles in right triangles. Concept Map ~ Key Points Example i) ABCD and PQRS are squares of sides 3 cm and • Two figures having same shapes (and not necessarily the same size) are called similar 5 cm respectively. The two squares are not figures. Two similar figures have the same shape congruent but they are similar as the ratios of but not necessarily the same size. their corresponding sides are equal  3 . • All squares with the same side length are congruent Scale factor = 3  5  and all equilateral triangles with the same side lengths are congruent. Any two equilateral triangles 5 are similar and any two squares are similar. i.e., the side of smaller square is 3 times the size of 5 • All congruent figures are similar but all similar figures need not be congruent. the larger square (or) the side of the large square is Similar Figures 1 = 5 times the side of the side of smaller square. • Two polygons of the same number of sides are 3 3 similar, if 5 i) All the corresponding angles are equal and We can write ABCD ∼ PQRS ii) All the corresponding sides are in the same ii) The circles having radii 1 cm and 1.5 cm are similar. ratio (or proportion). • In similar figures, the ratio of the corresponding Scale factor = 1 = 10 1.5 15 sides is referred to as the Sale Factor or Representative Fraction. • Similarity of two figures is denoted by the symbol “ ”. 45

6. Triangles R same ratio, then the line is parallel to the third side. This theorem is converse of Basic Proportionality 6.25 cm 7.5 cm Theorem. (BPT). 3 cm Criteria for Similarity of Triangles C o If in ∆ABC and ∆PQR, 2.5 cm o ∠A = ∠P , ∠B = ∠Q , ∠C = ∠R A=B B=C CA A 4 cm B P 10 cm Q PQ QR RP iii) The corresponding sides of the two triangles Then the two triangles are similar, ∆ABC~∆PQR have a ratio of • Theorem 6.3: Angle-Angle-Angle Similarity Criterion 4= 1 If in two triangles, corresponding angles are equal, 10 2.5 then their corresponding sides are in the same ratio (or proportion) and hence the two triangles 2.5 = 1 are similar. 6.25 2.5 Example: If in ∆ABC and ∆PQR, ∠A = ∠P , ∠B = ∠Q , ∠C = ∠R 3=1 7.5 2.5 then ∆ABC~∆PQR by Angle-Angle-Angle Similarity Criterion. The ratios of the corresponding sides are equal and ⇒ AB = BC = AC hence they are similar triangles. Scale factor = 1 PQ AR PR 2.5 A We can write ∆ABC~∆PQR P Similarity of Triangles B CQ R • Two triangles are similar if. • Angle-Angle Similarity Criterion o Their corresponding angles are equal and If two angles of one triangle are respectively equal o Their corresponding sides are in the same to two angles of another triangle, then the two triangles are similar. ratio (or proportion). Example: • If the corresponding angles of two triangles A are equal, then they are known as equiangular P triangles. • Thales Theorem The ratio of any two corresponding sides in two equiangular triangles is always the same. • Theorem 6.1: Basic Proportional Theorem If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. Example: In the figure, PQ || BC, by basic proportionality theorem, we have AP = AQ . PB QC A B CQ R If in ∆ABC and ∆PQR, ∠A = ∠P , ∠B = ∠Q then ∆ABC~∆PQR by Angle- PQ Angle similarity criteria (or) ∠B = ∠Q , ∠C = ∠R then ∆ABC~∆PQR by Angle- BC Angle similarity criteria (or) • Theorem 6.2: ∠A = ∠B , ∠C = ∠R then ∆ABC~∆PQR by Angle- If a line divides any two sides of a triangle in the Angle similarity criteria Then ∆ABC~∆PQR by Angle-Angle similarity criteria ⇒ AB = BC = AC PQ AR PR 46

6. Triangles • Theorem 6.4: Side-Side-Side Similarity Criterion A If in two triangles, sides of one triangle are P proportional (i.e., in the same ratio of) to the sides of the other triangle, then their corresponding B CQ R angles are equal and hence the two triangles are similar. If ∆ABC ~ ∆PQR then Example: ar ( ABC ) =  AB 2 =  BC 2 =  AC 2 A ar ( PQR )  PQ   QR   PR  P     B CQ R Pythagoras Theorem • Theorem 6.7 If in ∆ABC ~ ∆PQR, A=B B=C AC then If a perpendicular is drawn from the vertex of the PQ QR PR right angle to the hypotenuse then the triangles on both sides of the perpendicular are similar to the ∆ABC ~ ∆PQR by Side-Side-Side Similarity whole triangle and are similar to each other. Criterion. ⇒ ∠A = ∠P , ∠B = ∠Q , ∠C = ∠R A • Theorem 6.5: Side-Angle-Side Similarity Criterion B DC If one angle of a triangle is equal to one angle of another triangle and the sides including these Example: angles are proportional, then the two triangles are In ∆ABC, ∠A = 90° and AD ⊥ BC then similar. ∆BAC ∼ ∆BDA ⇒ BC = AB = AC Example: If in ∆ABC ~ ∆PQR BA BD AD ∠A = ∠P and AB = AC , then ∆ABC ~ ∆PQR by ∆BDA ∼ ∆ADC ⇒ AB = BD = AD PQ PR Side-Angle-Side Similarity Criterion (or) AC AD CD ∠B = ∠Q and AB = BC , then ∆ABC ~ ∆PQR by ∆BAC ∼ ∆ADC ⇒ BC = AB = AC PQ QR AC AD CD Side-Angle-Side Similarity Criterion (or) ∠C = ∠R and BC = AC , then ∆ABC ~ ∆PQR by • Theorem 6.8 In a right triangle, the square of the hypotenuse is QR PR equal to the sum of the squares of the other two sides. Side-Angle-Side Similarity Criterion. ⇒ ∠A = ∠P , ∠B = ∠Q , and � A=B B=C AC A PQ QR PR B C Areas of Similar Triangles Example: • The area of a geometrical figure is denoted by ar( ) Example: Area of ∆ABC is denoted as ar(ABC). • Theorem 6.6 The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 47

6. Triangles ∆ABC is a right triangle and BC is the hypotenuse, then we have BC2 = AB2 + AC2 • Baudhayan Theorem The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth). • Theorem 6.9 In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Example: If in ∅ABC , BC2 = AB2 + AC2 , then ∠BAC = 90° A B C Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Pre-requisites • Congruency of Triangles PS – 2 • Similar Figures • Similarity of Triangles PS – 3 PS – 4 Triangles • Criteria for Similarity of Triangles PS – 5 • Areas of Similar Triangles PS-6 • Pythagoras Theorem Self Evaluation Sheet Worksheet for “Triangles” Evaluation with Self Check ---- or Peer Check* 48

PRACTICE SHEET - 1 (PS-1) 1. State the different rules for Congruency of triangles. 2. State whether True or False. i)  An obtuse angle triangle has a hypotenuse. ii)  Two scalene triangles can be congruent. 3. Explain the mid-point theorem. 4. Explain Pythagoras theorem. PRACTICE SHEET - 2 (PS-2) 1. State any two differences between the congruency and similarity of two triangles. 2. Can two isosceles triangles be always similar? 3. Can an equilateral triangle of side 3 cm, a square of 3 cm side length and a circle of diameter of 3 cm be called similar figures? 4. Can pentagon and hexagon are similar figures? 5. In a trapezium ABCD, AB || CD. A line PQ parallel to BC passes through the point of intersection of the diagonals O, as shown. Prove that AQ = OB × OA . DP OD OC DP C O A QB 6. Prove midpoint theorem using Basic Proportionality Theorem. 49

PRACTICE SHEET - 3 (PS-3) 1. In a right angle triangle, a point O on the hypotenuse divides it in the ratio 1:2 as shown. If BPOQ forms a rectangle then show that OP × OQ = AP × CQ . A PO BQ C 2. ∆ABC is a right angled triangle with right angle at B. If P is any point on AB and PQ AC⏊AC then prove that ∆ABC - ∆AQP. Mention ratios of the corresponding sides of the two triangles. A Q P BC 3. If two triangles are similar, prove that the ratio of heights over corresponding bases is equal to ratio of the sides of triangle. 4. ∆ABC is similar to ∆PQR and the ratio of their sides AB = m . Find the ratio of their perimeter. PQ A P B CQ R 50

PRACTICE SHEET - 4 (PS-4) 1. Two isosceles triangles are similar with a scale factor of 5. What will be the ratio of their areas. 2. ∆ABC and ∆PQR are similar triangles such that the ratio of the areas of the triangles is 16 . Determine the 9 length of the sides of ∆PQR. A P 6 cm 5 cm B 4 cm C Q R 3. An old farmer has land in the shape of ∆ABC and he wants to divide it for three purposes. The farmer marks the points P, R and Q, S to trisect the sides AB and AC as shown. If the area of ∆ARS is m sq. units, then find the area of the following: A RS PQ i) BCQP  ii) PQSR B C 4. One sandwich was left in the plate and two brothers had to share it equally. The elder brother cuts the sandwich ABC along PQ so that the two portions are of equal area. If AB = 6 cm and BC = 6 cm, PQ ‖ BC, Find length of BP, PQ. Determine ratio PQ:BC. A (Use 2 = 1.41 ) Q P BC 51

PRACTICE SHEET - 5 (PS-5) 1. A flag pole is supported using as shown. One end of the rope is tied to the pole (point A) at height of 50m from the ground. The other end of the rope is fixed to the ground (point B) at a distance of from the center of the pole. Find the length of the rope used. A Pole Rope . OB 2. A line PQ is drawn parallel to the side of the right angled triangle OAB as shown such that OP = 2PA . Show that OR = RQ = QB. O R PQ 45° AB 3. In order to cross busy street a foot bridge was constructed as shown. The height of the bridge was 5 m its length was 12 m on either side of the street. The width of the street was 20 m. Calculate the increase in the travel distance to cross the street due to foot bridge. 12 m 5m 12 m 20 m 4. In the figure shown, Prove that ∆ABD ∼ ∆EFC if EF ⊥ BC . A F BD E C 5. Check which of the following combination of sides will form a right angle triangle? i) 2 3 , 4 5 , 6 7   ii)  13, 12, 5  iii)  6 , 18 , 3   iv)  7 , 5, 12 52

PRACTICE SHEET - 6 (PS-6) I. Choose the correct option. 1. ∆ABC ≈ ∆XYZ. The ratio of their areas is 144 : 121. What is the ratio of their corresponding sides? (A) 12 : 11 (B) 11: 12 (C) 1 : 2 (D) 2 : 1 2. The corresponding sides of two polygons are equal. Then the polygons are_________. (A) proportional (B) different (C) not similar (D) similar 3. In ∆ABC, X, Y and Z are the midpoints of AB, BC and AC respectively. Then ∆ABC : XYZ is _____. (A) 1 : 2 (B) 2 : 1 (C) 1 : 3 (D) 3 : 1 4. In ∆ABC, P and Q are the midpoints of AC and BC respectively. Then AB : PQ is ______________. (A) 2 : 3 (B) 1 : 4 (C) 2 : 1 (D) 1 : 2 5. ∆ABC ≅ ∆DEF. If AB = 7 cm, BC = 5 cm, AC = 8 cm and EF = 8 cm, the measure of EF + DE is _______. (A) 15 cm (B) 12 cm (C) 7.5 cm (D) 13 cm 6. If ∆PQR ≈ ∆XYZ and ∠ P = 45ºand ∠ R = 75º. Then ∠ Y is _____________. (A) 120º (B) 60º (C) 75º (D) 45º 7. In the given figure, if ED is parallel to CB, x =__________. (A) 3 cm (B) 6 cm (C) 2. 4 cm (D) 12 cm 8. What is the value of ‘x’ if AB is parallel to XY? (A) 2.4 (B) 3.4 (C) 3.6 (D) 3.2 9. If PQ = XZ, QR = XY and PR = YZ, then ____________. (A) ∆PQR ≈ ∆YZX (B) ∆PQR ≈ ∆XZY (C) ∆PQR ≈ ∆XYZ (D) ∆PRQ ≈ ∆XZY 10. The diagonals in a quadrilateral divide each other proportionately. Then it is a __________. (A) trapezium (B) rhombus (C) kite (D) square II. Short answer questions. 1. Find the value of ‘x’ from the given figure. 2. In the given figure, MN is parallel to RQ, MR = 17 and PQ = 32.2. What is the measure of PN? PM 6 3. A boy of height 120 cm is walking away from the base of a street light at a speed of 1.5 m/s. If the street light is 4.8 m high, find the length of the boy’s shadow after 6 seconds. 53

PRACTICE SHEET - 6 (PS-6) III. Long answer questions. 1. A ladder of height 6.5 m is placed against a wall in such a way that its foot is 2.5 m away from the wall. Find the height of the wall at which the ladder touches the wall. 2. The hypotenuse of a right angled triangle is 4 m more than thrice the shortest side and the third side is 1 m less than the hypotenuse. Find the sides of the triangle. 54

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins (1 Mark) 1. State Basic Proportionality Theorem. 2. Are rhombus and square similar figures? (1 Mark) 3. The lengths of three sides of a triangle are 26 , 18 and 12 units. Check if the triangle is a right angle triangle. (1 Mark) 4. If ∆ABC and ∆BPQ are similar triangles and the length of their sides are indicated in the figure. Find the lengths of the sides of the two triangles. (1 Mark) A 3 cm P 5 cm B 2 cm Q 4 cm C 55

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 5. ABCD is a rectangle as shown in the figure. QD and PQ are drawn perpendicular to AC and CD respectively. Show that ∆ADQ and ∆DQP (3 Marks) DP C Q B A 6. A point O lies between two parallel lines l and m such that its distance from line l is two times its distance from line m . Prove that any two lines drawn through point O will form triangles with lines l and m and the ratio of their areas is equal to 4.  (5 Marks) 7. In ∆ABC, D is a point on BC and DF || AB and DE || AC as shown. Prove that AB = AC . (3 Marks) AE CF A E F B DC 56

7. Coordinate Geometry Learning Outcome By the end of this lesson, a student will be able to: • Determine the distance between the points and find the geometry formed by points. • Determine coordinates of point between a two given points such that its distance from the points is in a given ratio. • Determine the area of triangle from the coordinates of the three vertices of the triangle. Concept Map Key Points O (0, 0) is OP = x2 + y2 • The distance of a point from y-axis is called its x-co- Section Formula ordinate or abscissa. The coordinates of a point on the x-axis are of the • The coordinates of the point P ( x, y) which divides form (x,0). The distance of a point from x-axis is called its y-co- the line segment joining the points A( x1, y1 ) and ordinate or ordinate. The coordinates of a point on the y-axis are of the B ( x2 , y2 ) internally in the ratio m1 : m2 is given by form (0,y). P ( x, y) =  m1x2 + m2 x1 , m1 y2 + m2 y1  • A linear equation in two variables of the form,  m1 + m2 m1 + m2  ax + by + c = 0 , (a, b are not simultaneously zero)   when represented graphically, gives a straight line. The above formula is known as section formula. Distance Formula If the ratio in which P divides AB is k:1 then the co- • The distance between the points P ( x1, y1 ) and Q ( x2 , y2 ) is ordinates of point P will be  kx2 + x1 , ky2 + y1  PQ = ( x2 − x1 )2 + ( y2 − y1 )2  k +1 k +1  The distance of a point P ( x, y) from the origin The midpoint of a line segment divides the line segment in the ratio of 1:1, therefore the coordinates of the midpoint P of the line joining A( x1, y1 ) and B ( x2 , y2 ) is P  x1 + x2 , y1 + y2  .  2 2  57

7. Coordinate Geometry • To find the area of polygon, divide it into triangular regions which have no common area and add the Area of Triangle areas of these regions. • Area of a triangle = 1 × base × height C 2 D height base AB • Area of a ∆ABC whose vertices are A( x1, y1 ) , Example: B ( x2 , y2 ) and C ( x3, y3 ) is equal to ar ( ABCD) = ar ( ABD) + ar ( BCD) ar ( ABC ) = 1  x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Locating points in a graph PS – 1 • Graph of a linear equation in two variables. of the form ax + by + c = 0 • Distance Formula PS – 2 Pair of Linear Equations in Two Variables • Section Formula PS – 3 • Area of Triangle PS – 4 Worksheet for “Coordinate Geometry” PS-5 Evaluation with Self Check or Peer Self Evaluation Sheet Check* 58

PRACTICE SHEET - 1 (PS-1) 1. Locate the following points in the graph i) (4,5)  ii) (3,–4)  iii) (–4,6)  iv) (–5,–5) vii) (6,0) v) (0,6)  vi) (–6,0)  vii) (0,–6)  Join the points (v), (vi), (vii) and (viii). 2. Draw the graph of y = x + 2 and x = y + 2 . PRACTICE SHEET - 2 (PS-2) 1. Find the distance between the following points: i)  (–3,–3), (4,3)  ii)  (7,–1), (3,7)  iii)  (1,2), (5,6)  iv)  (4,1), (2,–6) 2. Check whether the distance between the following points is equal: i)  (a,b), (–a,–b)  ii)  (–a,–b), (a,b)  iii)  (a,b), (–a,b)  iv)  (a,b), (a,–b) 3. A quadrilateral is formed by the A(–2,0), B(6,2), C(5,5) and D(–3,3). Find the nature of the quadrilateral. 4. A large ground is divided in the form of a grid and Rahul is standing at (6,5) and Ram is standing at (−4, −3). If they start to walk towards one another at the same speed, then where will they meet on x-axis and y axis? 5. Check if the points (4,3) , (−2,1) and (1, 2) form the three vertices of a triangle? 6. Find 4 points which are at a distance of 4 units from (2,3) . 7. Determine the type of the triangle if its vertices have the coordinates (1,1) , (6, 4) and (4,6) . 59

PRACTICE SHEET - 3 (PS-3) 1. Find the midpoint of the line joining the following points. i) (9, −2) , (−3,5)   ii) (0,0) , (−5,10)   iii) (−5, −5) , (5,5)  iv) (−5,0) , (5,0) v)  (0,5) , (0, −5)   iv) (−3,6) , (4, 2) 2. Find the coordinates of the points which divide the line joining the points A(–1,8) and B(9,6) into 5 equal parts. AP Q R SB 3. OA is a line segment such that the point (5,1) divides it in the ratio of 6:1. Find the coordinates of point A. 4. Find the ratio in which the x-axis and y-axis divide the line joining the points A(5, −4) and B (−6, 2) . Find the coordinates of the point of intersection of the line AB and the axes. 5. A line AB is to be extended to a point C such that BC = 2AB . If the coordinates of A and B are (4,1) and (6, −5) , find the coordinates of C. AB C PRACTICE SHEET - 4 (PS-4) 1. Find the area of the triangle formed by the points (−1,5) , (6,1) , (8,7). 2. Find the area of the triangle formed by the following points: i) (4,5) , (6,1) , (−2, 4)   ii) (3,5) , (7, −14) , (−6, −8) 3. Find the value of p such that (–3,5), (6,8) and (10,p) are collinear. 4. An engineer measures a piece of land ABCD and selects a portion of it PQR for building a giant statue and in the remaining area he want to develop parks. What is the area in which the parks would be developed? Y 6 B(6,6) 5 A(-3,3) 4 P(3,3) 3 2 1 X' -6 -5 -4 -3 -2 -1-1O 1 2 3 4 5 6 7 8 -2 Q(5,-2) -3 C(8,-4) R(0,-3) -4 -5 -6 -7 -8 D(0,-8) Y' 60

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. The coordinates of origin are_________. (A) (0, 4) (B) (0, 0) (C) (3, 0) (D) (1, 1) 2. Identify the formula to find the collinearity of three points A(x1, y1), B(x2, y2) and C(x3, y3). (A) 1  x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) (B) 1 x1 ( y1 − y2 ) + x2 ( y2 − y3 ) + x3 ( y3 − y1 ) 2 2 (C) 1  x1 ( y2 − y1 ) + x2 ( y3 − y2 ) + x3 ( y1 − y3 ) (D) 1  x3 ( y2 − y1 ) + x2 ( y3 − y2 ) + x1 ( y1 − y3 ) 2 2 3. The coordinates of the midpoint of (0, 6) and (0, -6) is _____. (A) (0, 4) (B) (4, 0) (C) (3, 0) (D) (0, 0) 4. The coordinates of the point P that divides A(x1, y1) and B (x2, y2) in the ratio k : 1 are _______. (A)  kx1 − x2 , ky1 − y2  (B)  kx2 − x1 , ky2 − y1   k +1 k +1   k +1 k +1  (C)  kx2 + x1 , ky2 + y1  (D)  kx1 + x2 , ky1 + y2   k +1 k +1   k +1 k +1  5. The coordinates of the point that divides (-2, 0) and (2, 0) in the ratio 1: 1 is ________. (A) (3, 6) (B) (4, 4) (C) (-1, -1) (D) (0, 0) 6. The distance PQ between the points P(x, y) and Q(a, b) is given by _________. (A) (B) (C) (D) 7. Identify the set of collinear points. (A) (0, 9), (3, 9), (-4, 9) (B) (0, 1), (1, 4), (-1, -9) (C) (4, 9), (-3, 5), (0, 0) (D) (0, 0), (2, 4), (-4, 7) 8. What is the distance of a point P(x, y) from the origin? (A) (B) (C) (D) 9. A point P divides the join of points Q and R in the ratio 3 : 1. Then ____________. (A) QP : QR = 3 : 1 (B) QR : QP = 3 : 1 (C) QP : PR = 3 : 1 (D) All of these 10. The point X divides the median AD of a ABC such that AX: XD = 2 : 1. Then X is the __________. (A) orthocentre (B) circumcentre (C) incentre (D) centroid II. Short answer questions. 1. Find the midpoint of the line segment joining X(-4, 8) and Y(-10, 12). 2. The point (x, y) is equidistant from the points (-14, 8) and (4, 6). Find the relation between x and y. 3. Do the points P(1, 7), Q(4, 2), R(–1, –1) and S(– 4, 4) form the vertices of a square? III. Long answer questions. 1. Determine if the points (2, -2), (2, 4) and (-1, 3) are collinear. 2. Find the area of the parallelogram with (0, 0), (2, 0), (3, 3) and (1, 2) as vertices. 61

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins (1 Mark) 1. For a given pair of points, how many points are equidistant from them? 2. Find the nature of the quadrilateral formed by the points P(4,1), Q(–1,4), R(–4,–1) and S(1,–4). (4 Marks) 3. Find the coordinates of the points of trisection of line segment joining the points (8, −6) and (4, −8) . (2 Marks) AB CD 4. Find the midpoint of the largest side of the triangle formed by the points A(−2, −2) , B (6, 2) and C (−1,6) . (2 Marks) 5. If (3, −4) is the midpoint of the line AB and the coordinates of A are given by (15,6) , then find the coo rdinate s of the point B. (2 Marks) 6. Find the area of the quadrilateral formed by the points A(−2,1) , B (8, 2) , C (1, 6) and D (−5,3) . (4 Marks) 62

8. Introduction to Trigonometry Learning Outcome By the end of this lesson, a student will be able to: • Calculate the trigonometric ratios in a given triangle. • Determine the value of the given trigonometric expressions using trigonometric ratios of definite angles. • Simplify the given trigonometric expressions using principles of complementary angles, trigonometric identities. Concept Map 0° 30° 45° 90° 60° Key Points ∠ABC = 90° is right aangle and side opposite to the right angle is called hypotenuse. • Pythagoras Theorem AC is hypotenuse of ∅ABC ∆ABC. In a right angle triangle, the square of the For ∠BAC or ∠A , AB is the adjacent side and BC is hypotenuse is equal to the sum of the squares of the opposite side. tEhxeamotphleer: twIno sidthees. right angle triangle ABC, For ∠BCA or ∠C , BC is the adjacent side and AB is ∠ABC = 90° the opposite side. AC is the side opposite to right angle (∠ABC = 90°) A and hence it is called the hypotenuse. By Pythagoras Theorem, AC2 = AB2 + BC2 A BC BC Trigonometric Ratios • The trigonometric ratios of an acute angle in a right • In a right angled triangle ABC, angle triangle express the relationship between the angle and the length of its sides. The Trigonometric ratios of the angle A of triangle 63

8. Introduction to Trigonometry ABC are defined as follows: Trigonometric Ratios of Some Specific Angles C θ 0ϒ 30ϒ 45ϒ 60ϒ 90ϒ hypotenuse sin 0 11 3 1 Opp. side 2 22 of A cos 1 31 1 0 A adj. side B 2 22 of A tan 0 11 Not 3 3 defined sine of ∠A = sin A = side opposite ∠A = BC hypotenuse AC Not 22 1 cosec defined 3 cosine of ∠A = cos A = side adjacent ∠A = AB 2 hypotenuse AC Not defined tangent of ∠A = tan A = side opposite to ∠A = BC sec 1 2 2 2 side adjacent to ∠A AB 3 cosecant of ∠A = cosec A = 1 Not 3 1 1 0 cot defined 3 sine of ∠A = hypotenuse = AC • Two angles are said to be complementary if their side opposite�∠A BC sum equals 90ϒ. secant of ∠A = sec A = 1 • In a right angle triangle, the two angles other the right angle are always complementary angles. cosine of ∠A Trigonometric Ratios of Complementary Angles = hypotenuse = AC ( 0° ≤ A ≤ 90° ) side adjacent to ∠A AB cotangent of ∠A = cot A = 1 sin A = cos (90° − A) cos A = sin (90° − A) sin (90° − A) = cos A cos (90° − A) = sin A tangent of ∠A tan A = cot (90° − A) cot A = tan (90° − A) = side adjacent to ∠A = AB tan (90° − A) = cot A cot (90° − A) = tan A side opposite to�∠A BC cosecA = sec(90° − A) sec A = cosec(90° − A) Also sec(90° − A) = cosecA t=an A s=in A sec A cosec(90° − A) = sec A cos A cosecA Trigonometric Identities • An equation involving trigonometric ratios of an c=ot A c=os A cosecA sin A sec A angle is called a trigonometric identity, if it true for all values of the angle(s) involved. • The values of the trigonometric ratios of an angle donot vary with the lengths of the sides of the cos2 A + sin2 A = 1 (0° ≤ A ≤ 90°) triangle. 1+ tan2 A = sec2 A (0° ≤ A < 90° ) • Correct Representation 1+ cot2 A = cosec2 A (0° < A ≤ 90°) Term Correct Wrong Representation representation (sin A)2 sin2 A sin A2 cos−1 A sec A (cos A)−1 applicable for Similar notations are all trigonometric ratios. 64

8. Introduction to Trigonometry Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Pythagoras theorem PS – 1 • Complementary angles Introduction to Trigonometry • Introduction PS – 2 • Trigonometric Ratios • Trigonometric ratios of Some Specific PS – 3 angles • Trigonometric ratios of complementary angles • Trigonometric Identities PS – 4 Worksheet for “Introduction to Trigonometry” PS-5 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 65

PRACTICE SHEET - 1 (PS-1) 1. Find the length of the hypotenuse if the lengths of the other two sides of the triangle are 8 cm and 16 cm. 2. If the length of the hypotenuse is 12 cm and one side of the right angle triangle is 4 cm, find the length of the third side. 3. Find the complementary angles for the following angles: i) 0ϒ  ii) 90ϒ  iii) 45ϒ  iv) 65ϒ v)  30ϒ  vi) 60ϒ vii)  15ϒ  viii)  89ϒ PRACTICE SHEET - 2 (PS-2) 1. State whether the following statements are true are false. Justify the answer: i)  sin u × cosecA = 1,if u = A ii)  tan u= hypotenuse adjacent side iii) sec u= tan u iv) tan u× cot u = 1 sin u 2. If the sides of right angle triangle are a, b, c and if a > b > c find values of the trigonometric ratios (sine, cosine and tangent) acute angles of the triangle. 3. In ∆ABD , AD = 2CD and BC = AD. Find the value of sin u, cos u and tan u. Also verify if ∆ABD is a right angle triangle? 4. In the given figure cos α = 1 find the value of sin β if CD = 2BC and AC = ED. 2 sin α 5. Can the sine of any acute angle in a right angle triangle be equal to 3? 6. If tan u= 3 then prove that 2 i) sin u+ cos u = 5  ii) sin u sec u-cos u cosec u =2.5   sin u− cos u (1-cotu ) iii)  (sin u+ cos u)2 = 25 cos2u 4 7. In ∆ABC, sin C = 4 . Determine the following: 5 i)  Perimeter of the triangle if BC = 5 cm. ii) tan B B AC 66

PRACTICE SHEET - 3 (PS-3) 1. A large land is in the shape of a rectangle in which the length of the diagonal is 30 m. The diagonal makes 30ϒ with the longer side of the rectangular plot. Determine the length of the sides of the rectangular plot. 2. A person travelling on an inclined road on a hill for 500 m realises that he reached a height of 250 m from the bottom of the hill. What is the angle of the road with respect to the ground? 3. Find the value of the following: i) csoins00°° + 2 tan 45° (3 − sec 60°) ii) tan 45° − sec2 30° + cosec30° sin 2 45° + cos2 30° iii) scions3300°°ccooss3600°°++ssiinn6300°°csoins 60° 60° iv) csoesce2c24455°°++tacno2t 2 30° 30° 4. Anil, Sunil and Suresh starts to climb a tree using ropes tied to its branches. Aman standing 15 m from the huge tree has viewed at 0o (horizontally) and saw Anil, who was struggling to climb. When he tilted his head up by 30o he found Sunil and he had tilted his head further by 30o and he saw Suresh. How far did the three friends climbed the tree by using ropes? 5. A stairway is to be planned to climb to a height of 15 m to reach the top of a building. It was decided to start the stairway 15 m from the wall of the building. What is the angle at which the stairway is to be built? 6. Simplify the following expressions: i) sin 32°sin 58° + cos 32°cos 58°   ii) 1+ cosec19° sec 71° iii) sin 24ϒsec 66 ϒ  iv) tan 75° + tan15° cot 75° + cot15° 7. In right angle triangle with right angle as shown, find the value of the following: i) sin ( A − B)  ii) cos (B + C ) iii)  sec( A − C )  iv) cosec( B + C ) C ab B c A 8. If A + B = 90° , express the following using a) only sin(�) b) only cos(� ) trigonometric ratios: i) tan A  ii) sec B 9. If sin 2 (A+ B +C) = 3 , sin 2 ( A+ B) = 1 and sin 2 A = 1 find the values of A, B, C. 4 2 4 10. Find the value of A, B if i) cos (2A + B) = 1 and sin ( A + 2B) = 1 2 ii) tan (3A + B) = 1 and tan ( A + B) = 1 3 67

PRACTICE SHEET - 4 (PS-4) 1. Prove that 1 = sin Acos A tan A + cot A 2. Prove that sin 35°cos 55° + cos 35°sin 55° = 1 3. Prove that sec A − tan A = cos A cos A + sin A 1+ 2sin Acos A 4. In a right angle triangle ABC with right angle at C, prove that sin Acos B + cos Asin B = 1 . 5. Prove that  cos A+ 1 2 +  sin A+ 1 2 −  tan A+ 1 2 = 5  cos A   sin A   tan A  6. Prove that 1− sec2 A + sec2 tan2 A = tan4 A 7. Prove that cosec4 A − cot4 A = 2cosec2 A −1 8. Prove that sec2 A + cosec2 A = (cot A + tan A)2 9. Prove that tan2 A + 2 tan4 A + tan6 A = sec6 A − sec4 A 68

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Which ratio represents sine of angle? Hypotenuse (A) Side adjacent to angle (B) Side opposite to angle Hypotenuse (C) Side opposite to angle (D) Hypotenuse Hypotenuse adjacent to angle Side 2. An identity among the following is _______. (A) cos2A + sin2A = 1 (B) cos2A – sin2A = 1 (C) cosec2A – sec2A = 1 (D) Both (A) and (C) 3. Tan2 45º is __________. (C) 0 (D) ∞ (A) 1 (B) 3 (C) tan A (D) cosec A 4. Sin (90º – A) = _____________. (C) tan2A – 1 (D) cosec2A – 1 (A) – cos A (B) cos A 5. For 0º < A ≤ 90º, cot2A = _____________. (A) sec2A – 1 (B) cos2A – 1 6. If sec A = , what is the value of cot A? (A) (B) (C) (D) 7. The value of sin 60º cos 60º is: (A) (B) (C) (D) 8. If tan = 1, what is the value of cos ? (A) (B) (C) (D) 9. If sin A = cos 2A, the value of A is ____________. (D) 30º (A) 0º (B) 45º (C) 60º (D) Tan A = 1 10. Which of these is correct? Cos A (A) Sec A = 1 (B) Sin A = 1 A (C) Cot A = 1 A Cos A Sec Sin II. Short answer questions. 1. Show that tan2 + tan4 = sec4 - sec2 . 2. If cot (A – 18º) = tan 2A, where 2A is acute, find the value of A. 3. If sec + tan = P, find the value of cosec . III. Long answer questions. . 1. If A, B and C are the three angles of a triangle, then show that 2. Prove that (cosec A – sin A)(sec A – cos A) . 69

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Fi)i nsdinthue= vcaolsuue of u such thiai)t sin u= tan u (1 Mark) 2. If tan u= 7 then show that 5 i) 1− sin2 u 25   ii)  sec2 u−1 = 1  (3 Marks) 1− cos2 = 49 tan2 u u 3. Find the value of i) s3ecta2n4350°°−−csoetc23405°° ii) 2ssienc26300°°−+3ctoasn226405°° iii) sseinc22 30° + sin 2 45° iv)  tan2 60° + sin2 60° + cos2 30°  (4 Marks) 30° + sec2 45° tan2 30° + sin 30° + cos2 60° 4. Simplify the following: (2 Marks) 70 i) sin 24° cos 66° + cos 24°sin 66°   tan 24° cot 66° − sec 24°cosec66° ii) cos 35° − cosec55°  sin 35° − sec 55° 5. If A + B = 90° , express the following using a using

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins a) only sin (� ) b) only sec(� ) trigonometric ratios: (2 Marks) i) cosec B  ii) cot B  (cosecA − sec A)2 (3 Marks) 6. Prove that (tan A + cot A)2 = 1− 2 cos Asin A 71

9. Some Applications of Trigonometry Learning Outcome • By the chapter of this lesson, a student will be able to determine the relation between the height of an object and the distance from an object using trigonometric ratios. Concept Map Key Points • Trigonometric Ratios of Some Specific Angles • The trigonometric ratios of an acute angle in a right θ 0° 30° 45° 60° 90° angle triangle express the relationship between the angle and the length of its sides. sin 0 1 13 1 2 22 The trigonometric ratios of the angle A of triangle ABC are defined as follows: cos 1 31 1 0 2 22 tan 0 1 1 3 Not de- 3 fined =sin A s=idheyoppopteonsiutsee∠A BC • The line of sight is the line drawn from the eye of AC an observer to a point on the object viewed by the observer. cos=ec A s=i1n A sidheyoppopteonsiutes=e∠A AC BC The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal =cos A s=idheyapdojtaecneunste∠A AB when the point being viewed is above the horizontal AC level, i.e., the case when we raise our head to look at the object. The angle of depression of a point on the object being viewed is the angle formed by the line sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed. s=ec A c=o1s A sidehaydpjoatceennutsteo=∠A AC AB =tan A ss=iiddeeaodpjpaocseintet ttoo∠∠AA BC AB c=ot A t=a1n A side adjacent ttoo=∠∠AA AB side opposite BC . Also t=an A cs=oinsAA sec A cosec A c=ot A cs=oinsAA cosecA . sec A 72

9. Some Applications of Trigonometry Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 CPHreA-rPeqTuEisRit:esQuadratic E•q uTaritgioonnoms etric Ratios PS – 2 Some Applications of Trig- PS-3 onometry • Height and Distances Self Evaluation Sheet Worksheet for “Some Applications of Trigonometry” Evaluation with Self Check ---- or Peer Check* 73

PRACTICE SHEET - 1 (PS-1) 1) In the right angled triangle with ∠C = 90°, AB = 6m, determine the lengths of AC, BC and the value of ∠BAC. 2) In the given figure AD = 3 m and AB = 5 m. Find the lengths of the following: i) AC ii) BC iii) BE iv) DE v) CE 3) In the given figure, AB = 4m, BC = 6m. Find the difference in the lengths of BE and CD. Also find the length of AE and AD. 74

PRACTICE SHEET - 2 (PS-2) 1) The angle of elevation to the top of the flag pole is 60°. If the height of the pole is 10 m, determine the distance from where the pole was observed. 2) The angle of depression to a point on the car parked at a distance of 20 m is 45° from an office building. Determine the location in the building from which the car was observed, if each floor of the building was 4 m. 3) A sliding (extendable) ladder is available which can be extended to a maximum length of 15 m and can be reduced to a length of 5 m. i) If the angle of elevation of an object is 30° 7) An astronaut from the International Space Station (ISS) sees at an angle of depression of from a point which is at a distance of 8 30° and 45° towards his right side to see two cities. If the ISS is at a height of 300 km when the m on the ground then find the length of observation was made, determine the distance ladder needed to reach the object from the between the cities. (Use 3 = 1.73 ) 8) In a village, a person who is 2 m tall is unable to specified point on the ground. get signal from cell phone tower situated near ii) If the angle at which the ladder is inclined his house. The person walks 30 m towards the tower to get a signal. If the angles of elevation to the ground is 45° and the ladder is fully of the tower from the house and from the final position are 30° and 45° respectively, determine stretched, then find the height that can be the height of the tower and the distance between the person and the tower when the reached by the ladder. signal was received. Is it possible to determine the distance travelled by the signal when the iii) The ladder is to be used at an angle 60° and signal was received? (Use 3 = 1.73 , 2 = 1.41 ) the object is found at a height of 10 m from 9) A plane is moving up in the air. Ravi sees the the ground. Determine the point at which plane at an angle of elevation of 30° and calls his friend Ganesh to show him the plane. Ganesh the ladder is to be placed on the ground and comes near Ravi after 2 minutes and when Ravi shows the plane to Ganesh, the angle of find the length to which the ladder is to be elevation was 60°. If the plane is moving at a speed of 800km/hr, determine the height at extended. (Use 3 = 1.73 , 2 = 1.41 ) which the plane is moving and the direction in 4) Raju is flying a kite using 100 m of thread. which the plane is moving. (Use 3 = 1.73 ) 10) Suresh is coming from the top floor of his The thread makes an angle of 30° with the building to the ground and sees his old friend horizontal. Due to strong wind, the kite began Mahesh at the gate of his building which was 80 m away from the lift. From the top floor the to rise and the thread made an angle of 60° angle of depression was 60° and after 5 minutes the angle of depression was 30°. Determine after some time. Determine the increase in the the speed at which the lift was moving. (Use height of the kite due to wind. 3 = 1.73 ) (Use 3 = 0.866 ) 2 5) Saritha and her friends went to climb a hill. They reached one hill of 60 m height and walked to the edge of it and looked at the huge water fall which was on the other side of the valley. Saritha had to look down at 30° to see the bottom of the water fall and had to look up at 45° to see the top of the water fall. Determine the height of the water fall and distance between Saritha and the water fall. 6) Four friends are standing at the windows of opposite buildings at A, B, P, Q respectively as shown in the figure. The angle of depression of P from A is 30° and the from Q the angle of elevation of A is 45° and the angle of depression to B is 30° If the distance between the buildings is 5 m, and B is at a height of 1 m from the ground level. Determine the heights of A, P and Q from the ground 1 level. (Use 3 = 0.577 ) 75

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. The angle formed between the eye and the object observed when the object is above the eye level is called ___________. (A) angle of depression (B) zero angle (C) angle of elevation (D) Both (A) and (C) 2. The angle formed between the eye and the object observed when the object is below the eye level is called ___________. (A) angle of depression (B) zero angle (C) angle of elevation (D) Both (A) and (B) 3. The ___________ can be found by using trigonometric ratios. (A) weight of an object (B) height of an object (C) distance between two objects (D) Both (B) and (C) 4. The tangent of the angle of elevation of the source of light when the height of the object and its shadow are equal, is ______________. 1 (A) 0 (B) 1 (C) 3 (D) 3 5. A pole 9 m high casts a 3 3 m long shadow on the ground. The sun’s elevation is __________. (A) 60° (B) 30° (C) 45° (D) 90° 6. The angle of elevation of the sun is 30°. What is the length of the shadow of an object of height 15 m? (A) 15 3 m (B) 5 3 m (C) 3 m (D) 15 m 1 7. A pole casts a shadow that is 303° times its height. The angle of elevation of the pole is __________. (A) 45° (B) (C) 90° (D) 60° 8. The foot of a ladder is 7 m away on the ground, from a tree of height 14 m. What is the angle of elevation? (A) 45° (B) 90° (C) 60° (D) 30° 9. The angle of depression of a point from the top of an electric pole 4 m tall, which casts a 4 3 m long shadow is ____________. (A) 90° (B) 60° (C) 45° (D) 30° 10. A ladder breaks at 20 m from the ground and falls making an angle of 30º with the ground. The distance between the top of the ladder and its foot is __________. (A) 3 3m (B) 20 3m (C) 20 m (D) 10 m II. Short answer questions. 1. A building casts a shadow of length 50 3 m when the sun is at an angle of elevation 30º. What is the height of the building? 2. From a point 25 m away from the foot of a tower, the angle of elevation of its top is 30º. What is the height of the tower? 3. The shadow of a tree is found to be 60 m longer when the angle of elevation of the Sun is 30º than when it is 60º. What is the height of the tree. III. Long answer questions. 1. The angle of elevation of the top of an electric pole from the foot of a pedestal is 45º and the angle of elevation of the top of the pedestal from the foot of the pole is 30º. What is the ratio of their heights? 2. Two buildings of the same height are on the opposite sides of a road 90 m wide. From a point between them on the road, the angles of elevation of their tops are 30º and 60º. Find the height of the buildings. 76

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1) What is angle of elevation and what is angle of depression? If the angle of elevation of object A is given and angle of depression of object B is given then determine the object at a higher level from the ground. (2 Marks) 2) A person is standing in the middle of the road, the angle of elevation to top of a building of height 60 m is 60° on one side of the road and on the other side of the road, the angle of elevation to the top of the building is 30°. Determine the distance between the buildings and the height of the second building. (4 Marks) 3) Ravi was standing in the second floor of the building which is at a height of 10 m from the ground. Ravi looked down on the street and saw two cars. Ravi is 1.5 m tall and the angle of depression for the first car was 30° and the angle of depression for the second car was 60°. Determine the distance between the two cars on the road. (3 Marks) 4) From an aeroplane flying 2 km above the ground, Ravi sees two cities and the angles of depressions for the two cities are 60° and 30° onto his left and right respectively. Determine the distance between the two cities on the ground. (3 Marks) 5) Standing over a water tank of 10 m height on the side of the road, Suresh sees the various landmarks of the village on one side of the water tank and the hill on the other side of the tank. The angle of depression to the water pump near the tank along the straight road was 30° and the angle of elevation to the top of the hill was 60°. Determine the height of the hill if the distance between the tank and the hill is 50 m. Also find the distance between the hill and the water pump. (Use 3 = 1.73 ) (3 Marks) 77

10. Circles Learning Outcome By the end of this lesson, a student will be able to: • Differentiate between tangent and secant to a circle. • Determine relation between the geometries formed by drawing tangents to circles. Concept Map Key Points • The tangent to a circle is a special case of the secant, when the two end points of its corresponding • Circle is a collection of points in a plane which are chords coincide. at a constant distance (radius) from a fixed point (center). • Theorem 10.1 The tangent at any point of a circle is perpendicular Center of Circle to the radius through the point of contact. Example: Circle Line l is tangent to the circle at point A. Let O be the center of the circle. • A line and circle in a plane can exist in three possible OA is the radius of the circle. ways as shown: OA ⊥ l l ll O A l A A B • For a tangent, the line containing the radius through the point of contact with a circle is sometimes Line l is a Non intersecting Line l is Tangent to cricle Line l is Secant to cricle called the normal to the circle at the point. In the figure, the line OA extended on both sides will be line with respect to circle at point A. through A, B called normal to the circle at A. When l is the secant, the line joining the points of Number of Tangents from a Point to a Circle intersection of line and circle is called the chord of • A point can be inside, outside or on the circle. the circle (AB is chord of the circle). Tangent to a Circle o There is no tangent to a circle passing through a • A tangent to a circle is a line that intersects the circle point inside the circle. at only one point, i.e., there is only one tangent at a point of the circle. o There is only one tangent to a circle passing through a point lying on the circle. o There are exactly two tangents to a circle through a point lying outside the circle. 78

10. Circles In the figure, PA and PB are the two tangents • Theorem 10.2 drawn from a point P which is outside the circle. The lengths drawn from an external point to a circle The length of the segment of the tangent from the are equal. external point (P) and the point of contact with the A circle is called the length of the tangent from the P point P to the circle. Length of the tangent is PA and PB when the point O is outside the circle. If the point lies on the circle, B the length of the tangent is zero. P is an external point; PA and PB are the two PA tangents to the circle. P PA = PB • OP is the angular bisector of the ∠APQ , i.e., X ∠OPA = ∠OPB Point inside circle - Point (P) on circle - B No tangents possible One tangent possible Point (P) outside circle - Two tangents possible Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Basic elements of circle – center, PS – 1 PS – 2 Circles radius, diameter, secant, tangent, PS – 3 chord, etc. PS-4 Worksheet for “Circles” • Tangent to a Circle Evaluation with Self Check or • Number of Tangents from a point on a Peer Check* Circle ---- Self Evaluation Sheet 79

PRACTICE SHEET - 1 (PS-1) 1. Name the following geometrical elements with respect to a circle whose center is at O. i) l  ii) AC  iii) OD  iv) BD  v) OF  vi) BF B AC l O F E 2. If a circle passes through the four vertices of a parallelogram, then it must be a rectangle or a square. Explain. 3. In the given figure, DE and AB are two equal chords of a circle. Find the value of the following. i) ∠OBA  ii) ∠ODE iii) ∠OBC E OC D 80° B A If DA = DE , find the value of ∠DOA . 80

PRACTICE SHEET - 2 (PS-2) 1. P, Q are two points on a plane. Point P is on the circumference of the circle and point Q lies inside the circle. State whether the following statements are true or false and justify. i) A line passing through P is always a secant to the circle. ii) A line passing through Q is always a tangent to the circle. 2. Draw two lines perpendicular to one another and secant to a circle. 3. Draw three lines as tangent to a circle so that they intersect to form a i) Triangle  ii) Quadrilateral 4. A line BA is drawn as a tangent to a circle at A. The lines OA and OB are drawn to complete the triangle. If the diameter of the circle is 6 cm and AB = 3 cm, find the value of BC. (Use 2 = 1.41) O C BA 5. Find the distance between a pair of parallel lines if they are tangent to a circle of diameter 8 cm. 81

PRACTICE SHEET - 3 (PS-3) 1. P, Q are two points on a circle whose center is where P is the point where side AB is tangent at O. Tangents are drawn from A passing to the circle of cream on the biscuits. through points P and Q. Prove that ∠QAP + ∠POQ = 180° . 6. ABCD is a quadrilateral in which a circle is Q inscribed with center at O, as shown in the figure. Find the value of ∠CDA of the O quadrilateral ABCD and determine if the line A DOB is a diagonal of the quadrilateral ABCD? C P 850 2. A line segment of length p units exists on a D plane. A circle of radius r (r < p) is drawn O through one of the end points. If tangents are 350 300 drawn from the external point of the line A B segment to the circle, show that the points lie at a distance of  p4 − p2  on either side of the line.  r2  7. A farmer had a large triangular piece of land 3. An isosceles triangle is drawn such that its whose sides are in the ratio of 8:9:10. Long ago sides are tangential to a circle whose center is he dug the well of 4 m diameter whose center at O. If AB = AC prove that R is the midpoint of is same as the center of the circle which was tangential to three sides of the land. Now the BC and point O lies on the line AR. farmer divides the land to his three sons by joining the center of the well and the vertices A of the land. Determine the ratio in which the farmer’s sons get the land. PQ O B RC 4. How many triangles can be constructed around a given circle such that the sides are tangent to the circle? 5. A biscuit manufacturing is making biscuits in the shape of parallelogram ABCD. In between two biscuits they put cream in circular shape so that the edges are tangential to the cream as shown in the figure. Determine if ABCD is a rhombus and find the radius of the cream if PB = 6 cm and radius of the cream OP = 3 cm, 82

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. The number of points common to a secant and a circle is/are ______. (A) 3 (B) 0 (C) 2 (D) 1 2. The point at which a tangent touches the circle is called ___. (D) point of contact (A) incentre (B) centroid (C) centre (D) Infinitely many 3. How many tangents can be drawn to a circle at a point on it? (A) 1 (B) 0 (C) 2 4. Identify the circle in which AB is a tangent. (A) (B) (C) (D) 5. The tangent at a point on a circle _______ the circle. (A) intersects (B) passes through the centre of (C) is parallel to (D) is perpendicular to 6. How many parallel tangents can a circle have? (A) 2 (B) 1 (C) 3 (D) 0 7. The lengths of tangents drawn from an external point to a circle are __________. ((A) always less than the radius of the circle (B) always equal (C) always greater than the radius of the circle (D) never equal 8. The angle between the tangent and the radius touching the point of contact is ______________. (A) an acute angle (B) a reflex angle (C) a right angle (D) a straight angle 9. TP and TQ are two tangents drawn to a circle with centre O from an external point T. Which of these is correct? (A) TP > TQ (B) TP < TQ (C) TP ⊥ TQ (D) TP = TQ 10. TP and TQ are two tangents drawn to a circle with centre O from an external point T. If PTQ = 60º, POQ = ___________ . (A) 60º (B) 120º (C) 30º (D) 90º II. Short answer questions. 1. Find the radius of the circle with centre O to which the tangent drawn from an external point is 12 cm and the distance of the point from the centre is 13 cm. 2. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 60º. Find the measure of POA. 3. Prove that if the line segment joining the points of contacts of two tangents is a diameter of a circle, the tangents are parallel to each other. III. Long answer questions. 1. In the given figure, A and B are the points of contact of the tangents PQ and XY. If AP = BX and APO = 30º, show that AOP + XOB = 4 APO. 2. In the given figure, AP and AQ are two tangents to a circle with centre O such that POQ = 70º. Find the measure of OPQ. 83

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins (1 Mark) 1. Whether normal to a circle is a secant or tangent to a circle? 2. Two points P, Q exist outside a circle whose center is at A. If the length of tangent from P and Q onto the circle are equal, then prove that OP = O. (1 Mark) 3 Two tangents are drawn from a point B to a circle of diameter 4 cm whose center is at O as shown in the figure. If AB = 5 cm, then find the value of OB and AC. (3 Marks) C O AB 4. A circle is inscribed in a triangle ABC as shown. Show that ∠CBA + ∠CAB = ∠POQ  (2 Marks) A P O BQ C 84

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 5. On a circle of diameter 6 cm, a rectangle ABCD is drawn that D is at the center of the circle and BC is tangential to the circle at C. Find the length of AP and BQ. (4 Marks) DC P Q 4 cm AB 6. A school has a large vacant land next to its building and they decided to make a cricket ground for the students. The land is in the shape of a triangle whose sides are 75 m, 70 m and 65 m. A large circular cricket ground was planned inside it. What will be the maximum diameter of the cricket ground possible?  (4 Marks) 85

11. Constructions Learning Outcome By the end of this chapter, a student will be able to: • Divide a line segment in the required ratio. • Draw a triangle similar to the given triangle. • Draw tangents to a circle from a given point. Concept Map Tangents to Circle Construction of Triangles Basic Proportionality Theorem Constructions Divide a line in a Construct a triangle Tangents to circle from given ratio similar to given triangle points outside it Key Points PA=QB QB=CR AC PR • If a line is drawn parallel to one side of a triangle to If in ∆ABC and ∆PQR, then intersect the other two sides in distinct points, then ∆ABC ~ ∆PQR the other two sides are divided in the same ratio. • Procedure to divide a line segment in a given ratio. Example: In the figure, PQ || BC, by basic Let AB be the given line to be divided in the ratio proportionality m:n theorem, we have AP = AQ . PB QC Let m = 3 and n = 2 A Method 1: Step 1: Line AB is located. PQ Step 2: Draw any line AX making an acute angle BC with AB. • If one angle of a triangle is equal to one angle of Step 3: Locate (m + n) =5 number of points another triangle and the sides including these angles are proportional, then the two triangles are A1 , A2 , A3 , A4 , A5 on the line AX such that similar. (SAS Similarity) =AA1 A=1 A2 A=2 A3 A=3 A4 A4 A5 . Example: A Step 4: Join BA5 P Step 5: Draw a line through the 3rd point (mth point) from A i.e., and parallel to BA5 such that it intersects the line AB at C B CQ R AC : BC = m : n 86