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TABLE OF CONTENTS 11 EXPONENTS 1 11.1 EXPONENTIAL FORM 1 11.2 LAWS OF EXPONENTS 6 11.3 EXPRESSING LARGE NUMBERS IN STANDARD FORM 12 12 QUADRILATERALS 14 12.1 QUADRILATERALS 14 12.2 TYPES OF QUADRILATERALS 18 13 AREA AND PERIMETER 25 13.1 INTRODUCTION 25 13.2 AREA OF A PARALLELOGRAM 27 13.3 AREA OF TRIANGLE 29 13.4 AREA OF RHOMBUS 34 13.5 CIRCUMFERENCE OF A CIRCLE 38 13.6 RECTANGULAR PATH 40 14 UNDERSTANDING 3D AND 2D SHAPES 42 14.1 INTRODUCTION 42 14.2 NETS OF 3–D SHAPES 46 14.3 DRAWING SOLIDS ON A FLAT SURFACE 55 14.4 VISUALISING SOLID OBJECTS 57 15 SYMMETRY 61 15.1 LINE SYMMETRY 61 15.2 ROTATIONAL SYMMETRY 68 15.3 LINE SYMMETRY AND ROTATIONAL SYMMETRY 72 74 PROJECT BASED QUESTIONS

CHAPTER 11 EXPONENTS EXERCISE 11.1 EXPONENTIAL FORM 11.1.1 Key Concepts i. When a number is multiplied by itself for many number of times (repeated multiplication) then we write it in exponential form. eg., 2 × 2 × 2 × 2 = 24 Here 2 is called base and 4 is called the exponent. 3 × 3 × 3 × 3 × 3 = 35 Here 3 is the base and 5 is the exponent. a.a.a.a . . . . . . ..a(m times) = am. Here ‘a’ is called the base and ‘m’ is called the exponent. ii. Very large numbers are easier to read, write and understand when expressed in exponential form. eg., 10000 = 104 8 × 8 × 8 × 8 × 8 × . . . . . . . . . . . . . × 8 (16 times) = 816. 11.1.2 Additional Questions Objective Questions . 1. [AS3] ‘10 raised to the power 3’ is (B) 103 (A) 310 (C)10 × 3 (D)None of these 2. [AS3] The exponential form of a × a × a × ........(k times) is . (A) ka (B) ka (C) ak (D)None of these EXERCISE 11.1. EXPONENTIAL FORM 1

3. [AS3] The expanded form of 65 is . (A) 6 × 6 × 6 × 6 × 6 (B) 5 × 5 × 5 × 5 × 5 × 5 (C)5 × 6 × 5 × 6 × 5 (D)6 × 6 × 6 × 6 × 6 × 6 4. [AS1] If a = 2 and b = 3 then the value of ab − ba is . (A) 1 (B) –1 (C) 2 (D) –2 5. [AS3] In 7x5, the base and exponent are respectively . (A) 7, x (B) 5, x (C) x, 5 (D)7x, 5 Very Short Answer Type Questions 6 [AS1] State true or false. [] (i) x 2 × y3 × z3 can be written as (x3× y2 × z2). [AS1] Fill in the blanks. (ii) a3× b2 × c is . (iii) (32)3 means . EXERCISE 11.1. EXPONENTIAL FORM 2

[AS1] Choose the correct answer. (B) (6)4 (iv) 74 means ____. (D)All of these (A) 7 × 7 (B) 4 × 4 × 4 × 4 (C)7 × 7 × 7 × 7 (D)None of these (v) 24 × 34 means _____. (A) (2 × 3)4 (C) 1296 7 [AS2] State true or false. (i) 52 × 22 means 25 × 4 [] 32 . [] 9 (ii) 60 × a0 × 70 is zero. [AS2] Fill in the blanks. . (iii) The value of 33 × 22 is (iv) xp × yp can be written as . [AS2] Choose the correct answer. (B) 0 (v) 2 −2 × 22 is _____. (D)None of these (A) 1 3 (C) 4 8 [AS3] Fill in the blanks. (i) 32 × 52 × 53 means . (ii) (−b) × (−b) × (−b) × (−b) is . EXERCISE 11.1. EXPONENTIAL FORM

(iii) 6 × 6 × 6 × 9 × 9 × 7 × 7 × 7 can be written as . [AS1] Choose the correct answer. (iv) 2 × 2 × 3 × 3 × 4 × 4 can be written as ____. (A) 22 × 32 × 42 (B) 2 × 3 × 4 (C)(2 × 2) − (3 × 3) − (4 × 4) (D)None of these (B) 33 (v) (−3) × (−3) × (−3) can be written as _______. (A) (−3)3 (C) (3)−3 (D)None of these 9 [AS1] State true or false. [] (i) 65 and 45 have the same base and the same exponent. [] [AS1] Fill in the blanks. 4 (ii) 3 5 and 36 have the same base. (iii) am × bm is . [AS1] Choose the correct answer. (B) Base (iv) p7 and q7 have the same ____. (D)None of these (A) Exponent (C)Base and exponent (v) [AS1] (2 × 3)3 is ____. (B) 216 (A) 16 (C) 6 (D) 36 10 [AS1] State true or false. (i) (2 × 3)4 can be written as (24 × 34) . EXERCISE 11.1. EXPONENTIAL FORM

Fill in the blanks. [AS1] (ii) x 2y3z4 can be written as . (iii) − 1 × 1 × 1 is . 2 2 2 [AS1] Choose the correct answer. (iv) 45 × 45 can be written as ____. (A) 35 × 45 (B) 3 × 5 (C)3 × 4 × 3 × 5 (D)None of these (v) (–a)4 can be written as ____. (A) (−a) × (+a) × (+a) × (−a) (B) (−a) × (−a) × (−a) × (−a) (C)(−a) × (a) × (−a) × (−a) (D)None of these Short Answer Type Questions 11(i) [AS1] Find the values of a) 24 × 33 b) 2 × 32 × 52 (ii) [AS1] Find the values of a) 23 × 34 × 52 b) 25 × 32 c) 52 × 23 × 72 EXERCISE 11.1. EXPONENTIAL FORM 5

EXERCISE 11.2 LAWS OF EXPONENTS 11.2.1 Key Concepts i. am.an = am+n ii. (am)n = amn iii. am.bm = (ab)m iv. am = an ⇒ m = n v. a−n = 1 an am vi. an = am−n ; if m > n = 1 ; if n > m an−m = 1 ; if m = n 7. a m = am b bm 8. a0 = 1 where a 0 11.2.2 Additional Questions Objective Questions . 1. [AS1] 53 × 54 = (B) 57 (A) 512 (D) 257 (C) 2512 2. [AS1] (72)3 = . (A) 76 (B) 75 (C) 723 (D) 77 3. [AS1] 8−2 = . (A) 16 (B) 10 (C) 1 (D) 1 16 64 EXERCISE 11.2. LAWS OF EXPONENTS 6

4. [AS1] If 2x × 25 = 28 then the value of x is . (A) 5 (B) 8 (D) 3 (C) 13 (B) 1 5. [AS1] The value of 9(40 + 50) is . (D) 18 (A) 0 (C) 9 Very Short Answer Type Questions [] 6 [AS1] State true or false. 7 (i) 5n = 125 then value of n is 3. [AS1] Fill in the blanks. (ii) If am = an then am = . an [AS1] Choose the correct answer. (iii) If x = 2 and y = 4 then the value of y x−y is ___. x (A) 4 (B) 1 4 (C) –4 (D) 2 3 (iv) The equivalent of 100 2 is ______. 9 (A) 9 3 100 2 (B) 1 3 100 2 9 (C) 3 × 3 × 3 100 10 10 (D) 100 × 100 × 100 9 9 9 EXERCISE 11.2. LAWS OF EXPONENTS

Short Answer Type Questions 7(i) [AS1] Simplify the following using am × an = am+n. a) 104 × 105 × 1014 b) 210 × (22)2 c) 32 × 92 (ii) [AS1] Find the values of 5 4, 55 and 59. Verify if 54 × 55 = 59. 8(i) [AS1] Simplify: (−3)5 3 (ii) [AS1] Simplify: (32)5 × (34)2 9(i) [AS1] Find the values of 23, 3 3 and 6.3 Verify if 3 3 = 63. 2 ×3 (ii) [AS1] a) Find the values of 32, 2 and 122. Verify if 2 42 = 122 . 4 3× b) Using a suitable law, write 23 × 93 in exponential form. 10(i) [AS1] Write the following with positive exponents. a) 4−5 b) 5−3 (ii) [AS1] Write the following with positive exponents. a) 3−8 . b) 4−12 c) 9 −4 EXERCISE 11.2. LAWS OF EXPONENTS 8

11(i) [AS1] Simplify the following using laws of exponents. a) 95 96 b) (−8)7 (−8)3 (ii) [AS1] If 7 2n+1 ÷ 49 = 73. Find the value of n. 12(i)[AS1] Simplify: a3b4 a2b8 (ii) [AS1] Simplify and write each of the following in exponential form. a) (−7)13 ÷ (−7)9 b) −3 7 −3 5 4 4 ÷ 13(i)[AS1] Simplify and write in exponential form: (−13)21 ÷ (−13)−13 (ii) [AS1] If 3 x + 3 ÷ 39 = 81 then ﬁnd x. 14(i)[AS1] If (−12) 7 ÷ (− 12)12 = ( − 12)x + 9 then ﬁnd x. (ii) [AS1] If 22x−9 ÷ 27 = 32 then ﬁnd x. 15(i)[AS1] Simplify and write in exponential form: (−4) −7 ÷ (−4)21 (ii) [AS1] If 52x − 9 ÷ 521 = 625 then ﬁnd x. 16(i)[AS1] Simplify and write in exponential form: 28 ÷ 323 (ii) [AS1] If 23x + 8 ÷ 4 = 128 then ﬁnd x. 17(i)[AS1] Simplify and write in exponential form: 25 4 ÷ 53 (ii) [AS1] Simplify: 124 × 52 × t8 103 × t4 18(i) [AS1] Simplify and write in exponential form: 3−2 ÷ 5−2 (ii) [AS1] Simplify and write in exponential form: x 12 × y24 × (23)4 y EXERCISE 11.2. LAWS OF EXPONENTS 9

19(i) [AS1] Simplify and write in exponential form: a) (25)3 ÷ 53 b) (81)5 ÷ (32)5 2 10 2 2 5 2 2n − 2 3 3 3 (ii) [AS1] If × = , ﬁnd the value of n. 20(i) [AS1] Simplify and write in exponential form. 5 6 × 52 2 2 (ii) [AS1] Simplify: 2 9 5 2 × 49 5 4 2 × 21(i) [AS1] Simplify and write in exponential form: a 5 × b10 b (ii) [AS1] If p = 2 2÷ 6 0 ﬁnd value of q3 q 3 7 , p. 22(i)[AS1] Simplify and write in exponential form: 3 5÷ 34 4 2 (ii) [AS1] Given 2 2n−9 = 4 0 Find the value of n. 7 7 . 23(i)[AS1] Simplify and write in exponential form: 75 ÷ 76 45 48 (ii) [AS1] Simplify and write in exponential form: −7 5 ÷ −7 3 8 8 24(i)[AS1] Simplify and write in exponential form: −7 9 × 75 8 2 (ii) [AS1] Find n such that 125 5 × 125 n = 5 18 8 8 2 . 25(i)[AS1] Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3. (ii) [AS1] Simplify using laws of exponents: (32)3 × 26 × 56 EXERCISE 11.2. LAWS OF EXPONENTS 10

26(i)[AS1] Simplify using laws of exponents: (72)3 ÷ 73 (ii) [AS1] Simplify using laws of exponents: (52)3 × 54 ÷ 57 27(i)[AS1] By what number should 7− 2 be multiplied so that the product is 343? (ii) [AS1] If 5x × 59 = 3125 then ﬁnd x. 28(i)[AS1] By what number should 5-4 be divided so that the quotient is 625? (ii) [AS1] If 2n − 5 × 5n − 4 = 5 then ﬁnd 'n'. 29(i)[AS1] If (2 2)n = (23)4 then ﬁnd 'n'. (ii) [AS1] If 25n−1 + 100 = 52n−1, ﬁnd the value of n. 30(i)[AS1] Find the value of 'm' so that m+1 (− 5 = (− 3)7 3) . (−3) × (ii) [AS1] Simplify: xa c xb a xc b xb xc xa × × 31(i)[AS1] Simplify using laws of exponents: 25 × 62 122 (ii) [AS1] If 32n + 7 × 3 − 12 = 243 then ﬁnd n. EXERCISE 11.2. LAWS OF EXPONENTS 11

EXERCISE 11.3 EXPRESSING LARGE NUMBERS IN STANDARD FORM 11.3.1 Key Concepts i. A number which is expressed as the product of the largest integer exponent of 10 and a decimal number between 1 and 10 is said to be in standard form. e.g., 6589 in standard form is 6.589 × 103 . 11.3.2 Additional Questions Objective Questions 1. [AS3] The standard form of 3908.78 is . (A) 39.0878 × 104 (B) 3.90878 × 103 (C)390.878 × 104 (D)3908.78 × 104 2. [AS1] The standard form of 723 × 109 is . (A) 7.23 × 1011 (B) 72.3 × 1011 (C)7.23 × 1012 (D)7.23 × 1013 3. [AS3] The usual form of 3.21 × 5 is . (B) 32,10,00,000 10 (D) 3,21,000 (A) 32,10,000 (C) 3,21,00,00,000 4. [AS3] The usual form of 3.5 × 103 is . (A) 350 (B) 3,500 (C) 35,000 (D) 35,00,000 EXERCISE 11.3. EXPRESSING LARGE NUMBERS IN STANDARD FORM 12

5. [AS3] The usual form of 9.325 × 1012 is . (A) 9,325,000,000,000 (B) 93,250,000 (C) 9,325,000 (D) 9,32,500 Very Short Answer Type Questions 6 [AS3] State true or false. (i) 1000 × 1000 is equal to 106 . [] [AS3] Fill in the blanks. (ii) 81 − 3 is equal to . 4 16 [AS3] Choose the correct answer. (iii) −27 can be represented as ______. 125 (A) −3 3 5 (B) −3 5 (C) 5 3 3 (D)None of these 243 −4 32 (iv) 5 = _____. (A) 61 (B) 18 18 61 (C) 81 (D) 16 61 81 (v) (125)5 is equal to _____. (B) 515 (A) 55 (D)None of these (C) 58 EXERCISE 11.3. EXPRESSING LARGE NUMBERS IN STANDARD FORM 13

CHAPTER 12 QUADRILATERALS EXERCISE 12.1 QUADRILATERALS 12.1.1 Key Concepts i. Quadrilateral: A closed ﬁgure bounded by four line segments is called a quadrilateral. ii. A quadrilateral divides a plane into three parts: interior of the quadrilateral, exterior of the quadrilateral and boundary of the quadrilateral. iii. A quadrilateral is said to be a convex quadrilateral if all line segments joining points in its interior also lie in its interior completely. e.g.,– BELT is a convex quadrilateral. iv. A quadrilateral is said to be a concave quadrilateral if all line segments joining any two points in its interior do not necessarily lie in its interior completely. In RING, the line segment ABdoes not lie completely in its interior. So, the quadrilateral RING is a concave quadrilateral. v. Sum of the interior angles of a quadrilateral is 360◦. EXERCISE 12.1. QUADRILATERALS 14

12.1.2 Additional Questions Objective Questions 1. [AS3] The sum of exterior angles of a quadrilateral is . (A) 180◦ (B) 270◦ (C) 360◦ (D) 720◦ 2. [AS3] The line segment joining the opposite vertices in a quadrilateral is called its . (A) Diagonal (B) Altitude (C) Median (D)None of these 3. [AS1] If the angles of a quadrilateral are 2x◦, (3x + 2)◦, (5x − 2)◦, 8x◦ then the angles are . (A) 30◦, 120◦, 170◦, 60◦ (B) 40◦, 62◦, 98◦, 160◦ (C)45◦, 89◦, 121◦, 105◦ (D)120◦, 150◦, 60◦, 30◦ 4. [AS1] If the three angles of a quadrilateral are 69◦, 98◦ and 129◦ then the fourth angle is . (A) 66◦ (B) 69◦ (C) 63◦ (D) 64◦ 5. [AS3] The number of diagonals that can be drawn in a quadrilateral is . (A) 1 (B) 2 (C) 3 (D) 4 EXERCISE 12.1. QUADRILATERALS 15

Very Short Answer Type Questions [ ] 6 [AS2] State true or false. ] (i) Rectangle is a convex quadrilateral. (ii) In a concave quadrilateral a line segment joining any two ponts will lie in its interior. [AS2] Fill in the blanks. [ (iii) In a trapezium ABCD, AB DC. If ∠D = x ◦ then ∠A = . (iv) is a quadrilateral. [AS2] Choose the correct answer. (v) is a ________. (A) Convex quadrilateral (B) Concave quadrilateral (C)Both convex and concave quadrilateral (D)None of these 7 [AS4] State true or false. (i) A parallelogram having all sides equal is called a rhombus. [ ] . [AS4] Fill in the blanks. (ii) A is a quadrilateral with one pair of parallel sides. (iii) If an angle of a parallelogram is a right angle, then it is necessarily a [AS4] Choose the correct answer. (iv) The consecutive angles of a parallelogram are ________. (A) Supplementary (B) Complementary (C)Right angles (D)None of these EXERCISE 12.1. QUADRILATERALS 16

(v) If a pair of opposite sides of a figure are equal and parallel then it is a . (A) Rectangle (B) Square (C) Rhombus (D) Parallelogram Correct Answer: D Short Answer Type Questions 8(i) [AS1] If three of the angles of a quadrilateral are 35◦, 95◦ and 125◦, ﬁnd the fourth angle. (ii) [AS1] If the adjacent angles of a parallelogram are in the ratio 5 : 4. Then ﬁnd all the four angles of the parallelogram. 9(i) [AS1] Write the adjacent side of AB and the adjacent angle of ∠A in the following ﬁgure. (ii) [AS1] Write the adjacent side of BC and the adjacent angle of ∠C in the following ﬁgure. EXERCISE 12.1. QUADRILATERALS 17

EXERCISE 12.2 TYPES OF QUADRILATERALS 12.2.1 Key Concepts i. A quadrilateral in which one pair of opposite sides are parallel is called a trapezium or trapeziod. In the given trapezium ABCD, AB is parallel to DC. ii. A trapezium in which non parallel sides are equal is called an isosceles trapezium or isosceles trapeziod. iii. A kite is a quadrilateral in which exactly two distinct pairs of sides are of the same length. In the kite ABCD, AB = BC and AD = CD. iv. A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram. In the quadrilateral ABCD, AB CD and AD BC. Hence, ABCD is aparallelogram. EXERCISE 12.2. TYPES OF QUADRILATERALS 18

v. In a parallelogram, • Opposite sides are parallel and equal [AB = CD and AD = BC]. • Diagonals bisect each other (AO = OC and BO = OD). • Opposite angles are equal (∠A = ∠C and ∠B = ∠D). • Adjacent angles are supplementary (∠A + ∠B = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 180◦). vi. A parallelogram in which adjacent sides are equal is called a rhombus. In quadrilateral ABCD, AB = BC = CD = DA and hence, ABCD is a rhombus. vii. In a rhombus, the diagonals bisect each other at right angles, i.e., AC ⊥ BD and AO = OC, BO = OD. viii. If one of the angles in a parallelogram is a right angle then it is a rectangle. In the quadrilateral ABCD, AB = DC and AD = BC, ∠A = ∠B = ∠C = ∠D = 90◦. So, ABCD isa rectangle. ix. In a rectangle, the diagonals are equal and bisect each other (AC = BD; AO = OC and BO = OD). x. A square is a rectangle with equal adjacent sides. In the quadrilateral ABCD, AB = BC = CD = DA, ∠A = ∠B = ∠C = ∠D = 90◦. So, ABCD is a square. xi. In a square, the diagonals are equal and bisect each other at right angles. EXERCISE 12.2. TYPES OF QUADRILATERALS 19

xii. Flow chart of family of quadrilaterals 12.2.2 Additional Questions Objective Questions . 1. [AS1] One of the adjacent angles of a parallelogram is 135◦. Then the other angle is . (A) 135◦ (B) 45◦ (C) 55◦ (D)None of these 2. [AS3] The quadrilateral among the following in which the diagonals are equal is a (A) Parallelogram (B) Rhombus (C) Rectangle (D) Kite 3. [AS1] If the lengths of two diagonals of a rhombus are 6 cm and 8 cm, the length of its side is . (A) 10 cm (B) 12 cm (C) 5 cm (D) 6 cm EXERCISE 12.2. TYPES OF QUADRILATERALS 20

4. [AS1] In a parallelogram PQRS, if PQ = 7 cm and QR = 9 cm then its perimeter is . (A) 32 cm (B) 16 cm (C)34 cm (D)18 cm 5. [AS3] In a rhombus if one of the angles is 90◦ then it becomes a . (A) Parallelogram (B) Rectangle (C) Trapezium (D) Square Very Short Answer Type Questions 6 [AS3] State true or false. [ ] (i) A kite has exactly two consecutive pairs of sides of equal length. . [AS4] Fill in the blanks. (ii) Diagonals necessarily bisect opposite angles in a . (iii) We get a rhombus by joining the mid points of the sides of a [AS3] Choose the correct answer. (iv) A quadrilateral with one pair of opposite sides parallel is called a _______. (A) Parallelogram (B) Rhombus (C) Rectangle (D) Trapezium (v) The ﬁgure formed by joining the mid points of the adjacent sides of square is a ______. (A) Square (B) Rhombus (C) Rectangle (D) Parallelogram EXERCISE 12.2. TYPES OF QUADRILATERALS 21

Short Answer Type Questions 7(i) [AS1] HELP is a parallelogram. Given that OE = 4 cm, where O is the point of intersection of the diagonals and HL is 5 cm more than PE, ﬁnd OH. (ii) [AS1] SHUB is a parallelogram. Given that OS = 7 cm, where O is the point of intersection of the diagonals and HB is 4 cm more than SU. Find BH + SU. 8(i) [AS1] In a rectangle PQRS, if PQ = 6 cm and PR = 10 cm then ﬁnd the perimeter of the rectangle. (ii) [AS1] Two adjacent angles of a parallelogram ABCD are in the ratio 2 : 3. Find all the angles of the parallelogram. 9(i) [AS1] The perimeter of a parallelogram is 80 m. If the longer side is 10 m greater than the shorter side, ﬁnd the lengths of the sides of the parallelogram. (ii) [AS1] The angles of a quadrilateral are in the ratio 3 : 5 :7 : 9. Find the angles. Long Answer Type Questions 10 [AS1] Find the perimeter of the parallelogram ABCD. EXERCISE 12.2. TYPES OF QUADRILATERALS 22

11 [AS1] In a parallelogram BEST, if ∠B = 60 ◦ Find the other angles. . 12 [AS1] ABCD is a parallelogram, given OA = 6 cm where O is the point of intersection of the diagonals and DB is 3 cm shorter than AC, ﬁnd OB. 13 [AS1] BASE is a rectangle. Its diagonals intersect at O. Find x, if OB = 5x + 1 and OE = 2x + 4. 14 [AS1] (i) Two adjacent sides of a parallelogram are in the ratio 4 : 5. The perimeter of the parallelogram is 54 cm. Find the length of each of its sides. (ii) In a trapezium RENT, RE is parallel to NT, ∠R = 110◦ and ∠E = 60◦. Find the remaining angles. 15 [AS2] The angles A, B, C and D of a quadrilateral ABCD are in the ratio 1 : 3 : 7 : 9. a) Find the measure of each angle. b) Is ABCD a trapezium? Why? c) Is ABCD a parallelogram? Why? EXERCISE 12.2. TYPES OF QUADRILATERALS 23

16 [AS1] In the given figure, HOPE is parallelogram. Find the values of x, y and z. 17 [AS1] In the given ﬁgure, RUNS is a parallelogram. Find x and y if their lengths are in cm. 18 [AS2] The diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Draw a rough ﬁgure to justify your answer. 19 [AS2] ABCD is a square with AC and BD as diagonals. Prove that AC and BD are equal. EXERCISE 12.2. TYPES OF QUADRILATERALS 24

CHAPTER 13 AREA AND PERIMETER EXERCISE 13.1 INTRODUCTION 13.1.1 Key Concepts i. The area of any plane ﬁgure is the region occupied by it on a plane. ii. The area of a rectangle is given by the product of its length (l) and breadth (b). i.e., A = l × b sq. units. iii. The area of a square is given by the square of its side (a) i.e., A = a 2sq. units. iv. The perimeter of a ﬁgure is the sum of the lengths of its all sides. For a rectangle it is twice the sum of its length and breadth. For a square, it is four times its side. 13.1.2 Additional Questions Objective Questions cm long. 1. [AS1] The area of a square is 289 sq. cm. Its side is (A) 14 (B) 15 (C) 16 (D) 17 2. [AS1] The perimeter of a square garden is 256 cm. The length of its side is cm. (A) 64 (B) 65 (C) 46 (D) 56 3. [AS1] If the area of a rectangle is 120 sq. cm and its length is 15 cm, its breadth is cm. (A) 7 (B) 8 (C) 9 (D) 10 EXERCISE 13.1. INTRODUCTION 25

4. [AS1] The length of a rectangular ﬁeld is 13 cm and its breadth is 10 cm. Its perimeter is cm. (A) 23 (B) 32 (C) 46 (D) 64 5. [AS3] The region occupied by a closed ﬁgure is called its . (A) Perimeter (B) Area (C) Length (D) Breadth Short Answer Type Questions 6(i) [AS1] A square of side of 6 cm and a rectangle of breadth 4 cm have the same area. What is the length of the rectangle? (ii) [AS1] Find the perimeter of the following rectangle. EXERCISE 13.1. INTRODUCTION 26

EXERCISE 13.2 AREA OF A PARALLELOGRAM 13.2.1 Key Concepts i. The area of a parallelogram is equal to the product of its base (b) and corresponding height (h) A = bh. ii. Any side of the parallelogram can be taken as its base. 13.2.2 Additional Questions Objective Questions 1. [AS1] The area of a parallelogram is 100 sq. cm. If the base is 25 cm, then the corresponding height is . (A) 4 cm (B) 5 cm (C)6 cm (D)7 cm 2. [AS1] The base of a parallelogram is twice its height. If its area is 512 sq. cm then the length of its base is . (A) 16 cm (B) 32 cm (C)48 cm (D)64 cm 3. [AS1] The area of a parallelogram with base 8 cm and altitude 4.5 cm is . (A) 36 cm2 (B) 32 cm2 (C)34 cm2 (D)36 m2 EXERCISE 13.2. AREA OF A PARALLELOGRAM 27

4. [AS1] The area of a parallelogram is 54 cm2 and its base is 12 cm. Its altitude is . (A) 6 cm (B) 4.5 cm (C)8 cm (D)9 cm 5. [AS1] The adjacent sides of a parallelogram are 10 m and 8 m. If the distance between the longer sides is 4 m then the distance between the shorter sides is . (A) 3 m (B) 5 m (C)7 m (D)9 m Short Answer Type Questions 6(i) [AS1] The base of a parallelogram is twice its height. If its area is 1250 sq. m, ﬁnd its base and height. (ii) [AS1] A parallelogram has a length of 15 cm and a height of 20 cm. It is divided into two congruent triangles. What is the area of each triangle? Long Answer Type Questions 7 [AS1] ABCD is a parallelogram with sides 10 cm and 8 cm. Find the measures of its base, height and area. 8 [AS1] Find the area of a parallelogram ABCD if AC = 18 cm, DF = 9 cm and BE = 7 cm. EXERCISE 13.2. AREA OF A PARALLELOGRAM 28

EXERCISE 13.3 AREA OF TRIANGLE 13.3.1 Key Concepts i. The area of a triangle is equal to half the product of its base and height. A = 1 bh 2 ii. Area of a triangle = Half the area of a parallelogram with the same base and height. iii. Area of a right angled triangle = ab where ‘a’ and ‘b’ are the lengths of the perpendicular sides of 2 the triangle. 13.3.2 Additional Questions Objective Questions 1. [AS1] The area of a triangle is 28 sq. cm. and its base is 14 cm. Its height is . (A) 3 cm (B) 4 cm (C)5 cm (D)6 cm 2. [AS1] The base and height of a triangle are in the ratio 3 : 2. Its area is 75 sq. m. Then base and height are ____. (A) 12 m and 10 m (B) 13 m and 15 m (C)15 m and 10 m (D)16 m and 12 m 3. [AS1] The base of a triangle is twice its height, and its area is 49 sq. cm. Its base and height are . (A) 14 cm and 7 cm (B) 12 cm and 6 cm (C)10 cm and 5 cm (D)16 cm and 8 cm EXERCISE 13.3. AREA OF TRIANGLE 29

4. [AS1] In a right angled triangle ABC, right angled at B, AB = 6 cm and AC = 10 cm. The area of triangle ABC is . (A) 28 sq. cm (B) 27 sq. cm (C)24 sq. cm (D)26 sq. cm 5. [AS1] The area of triangle is equal to the area of a square whose side is 10 cm. If the base of the triangle is 25 cm, then its height is . (A) 2 cm (B) 3 cm (C)8 cm (D)5 cm Short Answer Type Questions 6(i) [AS1] ∆ ABC is right angled at A, AD ⊥ BC, AB = 5 cm, BC = 13 cm and AC = 12 cm. Find the area of ∆ ABC. Also, ﬁnd the length of AD. (ii) [AS1] Find the perimeter and the area of a right angled triangle whose sides containing right angle are 5 cm and 12 cm and its hypotenuse is 13 cm. Long Answer Type Questions 7 [AS1] Find the height of a triangle whose area is 450 sq. cm. and base is equal to 15 cm. 8 [AS1] The area of a triangle is equal to the area of a rectangle whose length and breadth are 18 cm and 12 cm respectively. Calculate the height of the triangle if its base is 40 cm. 9 [AS1] The area of a triangle, whose base and the corresponding altitude are 15 cm and 7 cm is equal to the area of a right triangle one of whose sides containing the right angle is 10.5 cm. Findthe other side containing the right angle. 10 [AS1] Triangle ABC is isosceles with AB = AC = 7.5 cm and BC = 9 cm. The height from A to BC i.e., AD is 6 cm. Find the area of triangle ABC. What will be the height from C to AB? EXERCISE 13.3. AREA OF TRIANGLE 30

11 [AS1] Triangle ABC is right angled at A. AD is perpendicular to BC. If AB = 9 cm, BC = 15 cm and AC = 12 cm, ﬁnd the area of triangle ABC. Also, ﬁnd the length of AD. 12 [AS1] a) Find the altitude of a triangle whose base is 20 cm and area is 150 sq. cm. b) The area of a triangle is equal to that of a square whose side measures 60 metres. Find theside of the triangle whose corresponding altitude is 90 metres. 13 [AS1] a) Find the altitude of a triangle whose area is 42 sq. cm. and base is 12 cm. b) The area of a triangle is 50 sq. cm. If the altitude is 8 cm, what is its base? 14 [AS1] Calculate the areas of the following triangles and check if they are the same. 15 [AS1] In the following ﬁgure, ﬁnd the area of the shaded portion. EXERCISE 13.3. AREA OF TRIANGLE 31

16 [AS1] Find the area of the shaded portion in the given figure. 17 [AS1] ABCD is a rectangle with dimensions 32 m by 18 m. ADE is a triangle such that EF ⊥ AD and EF = 14 m. Calculate the area of the shaded region. 18 [AS1] In the ﬁgure ABCD, ﬁnd the area of the shaded region. EXERCISE 13.3. AREA OF TRIANGLE 32

19 [AS1] Find the area of the shaded region. 20 [AS1] Find the area of the shaded region. 21 [AS1] Find the area of the shaded region. EXERCISE 13.3. AREA OF TRIANGLE 33

EXERCISE 13.4 AREA OF RHOMBUS 13.4.1 Key Concepts i. The area of a rhombus is equal to half the product of its diagonals. A = 1 d1d2 2 13.4.2 Additional Questions Objective Questions 1. [AS1] The area of a rhombus is 144 sq. cm. and one of its diagonals is double the other. The length of the longer diagonal is . (A) 12 cm (B) 16 cm (C)18 cm (D)24 cm 2. [AS1] The lengths of the diagonals of a rhombus are 36 cm and 22.5 cm. Its area is . (A) 810 cm2 (B) 405 cm2 (C)202.5 cm2 (D)1620 cm2 3. [AS1] The length of a diagonal of a rhombus is 16 cm. Its area is 96 sq. cm. Then the length of the other diagonal is . (A) 6 cm (B) 8 cm (C)12 cm (D)18 cm EXERCISE 13.4. AREA OF RHOMBUS 34

4. [AS1] The lengths of the diagonals of a rhombus are 8 cm and 14 cm. The area of one of the four triangles formed by the diagonals is . (A) 12 sq.cm (B) 8 sq.cm (C)13 sq.cm (D)14 sq.cm 5. [AS1] The area of a rhombus whose side is 15 cm and length of one diagonal is 24 cm is . ] (A) 216 sq. cm (B) 348 sq. cm (C)484 sq. cm (D)684 sq. cm Very Short Answer Type Questions 6 [AS1] State true or false. 2 × area . (i) In a rhombus, d2 = [ d1 [AS1] Fill in the blanks. (ii) 1 × d1 × 40 cm = 40. Then the value of d1 is 2 cm. 2 (iii) The length of BD is . [AS1] Choose the correct answer. (iv) The area of a rhombus is 60 cm and one of the diagonals is 12 cm. The other diagonal is . (A) 6 cm (B) 5 cm (C)10 cm (D)12 cm EXERCISE 13.4. AREA OF RHOMBUS 35

(v) The area of the rhombus is _____. (B) 6 sq. cm (A) 20 sq. cm (D)24 sq. cm (C)12 sq. cm Short Answer Type Questions 7(i) [AS1] If the length of a diagonal of a rhombus whose area is 300 sq. cm is 20 cm, find the length of the second diagonal. (ii) [AS1] If the side of a rhombus is 30 cm and the length of one of its diagonals is 48 cm, findthe length of the second diagonal. 8(i) [AS1] The diagonals of a rhombus are 15 cm and 12 cm respectively. Find its area. (ii) [AS1] The area of a rhombus is 150 sq. cm. If one of its diagonals is 3 times the other, find the lengths of the diagonals. Long Answer Type Questions 9 [AS1] The length of a diagonal of a rhombus is 20 cm. If its area is 200 sq. cm, ﬁnd the length of the second diagonal. 10 [AS1] a) The area of a rhombus is 336 sq. cm and the length of one of its diagonals is 28 cm. Find the length of the other diagonal. b) The side of a rhombus is 15 cm and length of one of its diagonals is 24 cm. Find the length of the other diagonal. EXERCISE 13.4. AREA OF RHOMBUS 36

11 [AS1] a) The length of a diagonal of a rhombus is 9 cm. If its area is 54 sq. cm, ﬁnd the length of its second diagonal. b) The length of a diagonal of a rhombus is 38 cm. If its area is 874 sq. cm, and the length of its second diagonal. 12 [AS4] The ﬂoor of a building consists of 3000 rhombus shaped tiles. If the diagonals of each tile are 45 cm and 30 cm, ﬁnd the total cost of polishing the ﬂoor at Rs. 2.25 per squaremetre. EXERCISE 13.4. AREA OF RHOMBUS 37

EXERCISE 13.5 CIRCUMFERENCE OF A CIRCLE 13.5.1 Key Concepts i. The circumference of a circle is given by 2πr or πd where π = 22 or 3.14. 7 13.5.2 Additional Questions Objective Questions 1. [AS1] The circumference of a circle whose radius is 21 cm is . (A) 123 cm (B) 128 cm (C)132 cm (D)143 cm 2. [AS1] The circumference of a circle whose diameter is 14 cm is . (A) 44 cm (B) 35 cm (C)48 cm (D)56 cm 3. [AS1] If the circumference of a circle is 220 cm then its radius is . (A) 36 cm (B) 37 cm (C)38 cm (D)35 cm 4. [AS1] If the circumference of a circle is 264 cm then its diameter is . (A) 12 cm (B) 42 cm (C)84 cm (D)82 cm EXERCISE 13.5. CIRCUMFERENCE OF A CIRCLE 38

5. [AS1] The number of times a wheel of radius 28 cm rotates to move through 704 cm is . (A) 3 (B) 4 (C) 5 (D) 6 Short Answer Type Questions 6(i) [AS1] Find the circumference of a circle of diameter 14 cm. (ii) [AS1] The circumference of a circle is 308 cm. Find the area of the circle. Long Answer Type Questions 7 [AS1] If the circumference of a circle is 352 cm, find its radius. T ake π = 22 . 7 8 [AS4] A road roller makes 200 rotations to cover 2200 m. Find its radius. EXERCISE 13.5. CIRCUMFERENCE OF A CIRCLE 39

EXERCISE 13.6 RECTANGULAR PATH 13.6.1 Key Concepts i. The area of a rectangle is given by the product of its length and breadth (l b). 13.6.2 Additional Questions Objective Questions 1. [AS1] A scenery is painted on a cardboard 19 cm long and 14 cm wide, such that there is a margin of 1.5 cm along each of its sides. The total area of the margin is . (A) 80 sq. cm (B) 88 sq. cm (C)90 sq. cm (D)108 sq. cm 2. [AS4] A path of width 2 metres runs around and outside a rectangular plot of length 25 metres and width 20 metres. The area of the path is . (A) 169 sq. cm (B) 196 sq. cm (C)200 sq. cm (D)180 sq. cm 3. [AS4] A path of width 2 metres, runs around and outside a rectangular plot of width 12 metres. If the area of the path is 24 square metres then the length of the rectangular plot is . (A) 12 metres (B) 13 metres (C)14 metres (D)15 metres 4. [AS4] A path of 3 metres runs around and outside a rectangular plot of length 26 metres and width 22 metres. The area of the path is . (A) 785 square metres (B) 257 square metres (C)125 square metres (D)324 square metres EXERCISE 13.6. RECTANGULAR PATH 40

5. [AS4] A path of 1 metre runs around and outside a rectangular plot of length 20 metres. If the area of the path is 74 square metres then the width of the rectangular plot is . (A) 9 metres (B) 5 metres (C)10 metres (D)15 metres Long Answer Type Questions 6 [AS5] A verandah 2 m wide is constructed outside all around a room of dimensions as given in the ﬁgure. Find the area of the verandah. 7 [AS4] A path 5 m wide runs outside around a square park of side 100 m. Find the area of the path. Also, ﬁnd the cost of cementing it at the rate of Rs. 200 per 10 sq. m. EXERCISE 13.6. RECTANGULAR PATH 41

CHAPTER 14 UNDERSTANDING 3D AND 2D SHAPES EXERCISE 14.1 INTRODUCTION 14.1.1 Key Concepts i. We see different two–dimensional and three–dimensional shapes around us. ii. We can identify geometric shapes in 2 and 3 dimensional shapes. iii. The 2 and 3 dimensional shapes are a combination of geometric shapes. iv. We can identify the faces, edges and vertices of 3 dimensional shapes. 14.1.2 Additional Questions Objective Questions . 1. [AS3] The number of edges of a cube is (A) 10 (B) 12 (C) 13 (D) 14 2. [AS3] The number of faces of a cuboid is . (A) 10 (B) 12 (C) 6 (D) 8 3. [AS3] The number of vertices of a cube is . (A) 4 (B) 5 (C) 7 (D) 8 EXERCISE 14.1. INTRODUCTION 42

4. [AS3] The number of faces of a square pyramid is . (A) 3 (B) 4 (C) 5 (D) 6 5. [AS3] The number of dimensions of a solid is . (A) 1 (B) 2 (C) 3 (D) 4 Very Short Answer Type Questions 6. [AS4] Match the real life objects with their basic 3D shapes. Column A Column B i. Cube a. Battery cell ii. Cylinder b. Dice iii. Sphere c. Brick iv. Cuboid d. Ice cream v. Cone e. Ball EXERCISE 14.1. INTRODUCTION 43

Long Answer Type Questions 7 [AS5] Identify and state the number of faces, edges and vertices of the ﬁgures given. S.No 3 –D ﬁgures No. of No. of No. of i. vertices edges faces ii. iii. iv. EXERCISE 14.1. INTRODUCTION 44

8 [AS5] Identify and state the number of faces, edges and vertices of the ﬁgures given. S.No 3 –D ﬁgures No. of No. of No. of i. Sphere vertices edges faces ii. Triangular prism iii. Rectangular pyramid iv. Square pyramid EXERCISE 14.1. INTRODUCTION 45

EXERCISE 14.2 NETS OF 3-D SHAPES 14.2.1 Key Concepts i. A net is a sort of skeleton – outline in 2–D, which, when folded, results in a 3–D shape. ii. A 3-D shape can have more than one net based on the way we cut it. Example: iii. 3–D shapes can be visualized by drawing their nets on 2–D surfaces. EXERCISE 14.2. NETS OF 3–D SHAPES 46

14.2.2 Additional Questions 47 Objective Questions 1. [AS5] Identify the net of a cone. (A) (B) (C) (D) EXERCISE 14.2. NETS OF 3–D SHAPES

2. [AS5] Identify the net of a cube. (A) (B) (C) (D) EXERCISE 14.2. NETS OF 3–D SHAPES 48

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