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TABLE OF CONTENTS 10 PERIMETER AND AREA 1 10.1 PERIMETER 1 10.2 AREA 5 11 RATIO AND PROPORTION 9 11.1 COMPARING QUANTITIES WITH DIFFERENT UNITS 9 11.2 RATIO IN DIFFERENT SITUATIONS 12 11.3 DIVISION OF A GIVEN QUANTITY IN A GIVEN RATIO 16 11.4 PROPORTION AND UNITARY METHOD 18 12 SYMMETRY 21 12.1 LINE SYMMETRY 21 12.2 MULTIPLE LINES OF SYMMETRY 26 13 PRACTICAL GEOMETRY 30 13.1 A LINE SEGMENT 30 13.2 CONSTRUCTION OF A CIRCLE 32 13.3 PERPENDICULARS 34 13.4 CONSTRUCTION OF ANGLES USING PROTRACTOR 37 13.5 CONSTRUCTING ANGLES OF SPECIAL MEASURES 39 14 UNDERSTANDING 3D AND 2D SHAPES 41 14.1 3–D SHAPES 41 14.2 POLYGONS 48 PROJECT BASED QUESTIONS 54

CHAPTER 10 PERIMETER AND AREA EXERCISE 10.1 PERIMETER 10.1.1 Key Concepts i. Perimeter is the distance covered along the boundary forming a closed ﬁgure when you go around the ﬁgure once. ii. a. Perimeter of a rectangle = 2 × (length + breadth) b. Perimeter of a square = 4 × length o f its side c. Perimeter of an equilateral triangle = 3 × length o f any side iii. a. Figures in which all sides and angles are equal are called regular closed ﬁgures. b. The perimeter of a regular ﬁgure is equal to the number of sides multiplied by the length of each side. 10.1.2 Additional Questions Objective Questions 1. [AS1] The perimeter of a square with side 8 cm is ______. (A) 2 cm (B) 32 cm (C)64 cm (D)16 cm 2. [AS1] The length of the side of a square if the perimeter is 48 cm is ______. (A) 8 cm (B) 4 cm (C)12 cm (D)16 cm EXERCISE 10.1. PERIMETER 1

3. [AS4] A wire in the shape of an equilateral triangle of side 10 cm is bent into a regular pentagon. Each side of the pentagon so formed is _______. (A) 8 cm (B) 4 cm (C)6 cm (D)12 cm 4. [AS1] The perimeter of an equilateral triangle with side 5 cm is ______. (A) 15 cm (B) 20 cm (C)25 cm (D)60 cm 5. [AS3] The perimeter of a rectangle of length ’l’ units and breadth ’b’ units is units. (A) 2(l –b) (B) lb (C) l (D)2(l + b) b Very Short Answer Type Questions . 6 Fill in the blanks. (i) [AS3] The length of the boundary of a ﬁgure is called its (ii) [AS3] The perimeter of an equilateral triangle . (iii) [AS1] If the perimeter of a regular pentagon is 10 cm, its side is . . (iv) [AS2] If the side of a square is doubled, its perimeter is (v) [AS3] Side of square = perimeter. Short Answer Type Questions 7 [AS1] Find the perimeter of following: (i) A table with sides 30 cm, 15 cm, 30 cm, 15 cm. (ii) Your mathematics textbook. 8(i) [AS4] a) A rectangular ﬁeld is 50 m by 40 m. Mahesh goes ten times around it. What is the distance covered by him? b) Find the breadth of the rectangular ﬁeld whose length is 70 cm and perimeter is 200 cm. EXERCISE 10.1. PERIMETER 2

(ii) [AS4] The length of a rectangular ﬁeld is twice its breadth. A man jogged around it 5 times and covered a distance of 3 km. What is the length of the ﬁeld? 9 [AS4] A rectangular park of sides 100 m and 70 m has to be fenced with a barbed wire. The cost of the wire is Rs. 20 per metre. Find the length of the wire required for fencing and also the cost of fencing the park. Long Answer Type Questions 10 [AS5] Find the perimeters of the following ﬁgures. (i) (ii) 11 [AS1] Find the length of the side of: (i) a square if its perimeter is 48 cm. (ii) an equilateral triangle if its perimeter is 36 cm. (iii) a regular pentagon if its perimeter is 65 cm. 12 [AS4] A wire in the shape of an equilateral triangle of side 16 cm is bent into a regular octagon. Find the length of each side of the octagon. EXERCISE 10.1. PERIMETER 3

13 [AS1] (i) Find the perimeter of a rectangle which is 54 m long and 36 m wide. (ii) Find the length of a rectangle whose perimeter is 80 cm and breadth is 12 cm. (iii) Find the breadth of a rectangle whose perimeter is 120 m and length is 36 m. 14 [AS4] A piece of string is 120 cm long. What will be the length of each side if the string is bent to form: (i) a square? (ii) an equilateral triangle? (iii) a regular pentagon? (iv) a regular hexagon? 15 [AS4] Mr. Varma has an orchard of length and breadth 280 m and 200 m respectively. He wants to fence it with 4 rounds of barbed wire. Find the cost of fencing at Rs. 35 per metre. EXERCISE 10.1. PERIMETER 4

EXERCISE 10.2 AREA 10.2.1 Key Concepts i. The amount of surface enclosed by a closed ﬁgure is called its area. ii. To calculate the area of a ﬁgure using a squared paper, the following conventions are adopted. a. Ignore portions of the area that are less than half a square. b. Count more than half a square in the region as one square. c. Take the exact half square as ½ sq. unit. iii. Area of a rectangle = length × breadth iv. Area of a square = side × side v. The area of a square is more than the area of a rectangle having the same perimeter as that of a square. 10.2.2 Additional Questions Objective Questions 1. [AS1] The area of a square with perimeter 28 cm is _____. (A) 7 cm2 (B) 49 cm 2 (C)784 cm 2 (D)196 cm2 2. [AS1] The area of a square whose perimeter is equal to perimeter of a rectangle with length 10 cm and breadth 6 cm is ______. (A) 64 cm2 (B) 8 cm 2 (C)16 cm 2 (D)32 cm2 3. [AS2] If the length of a rectangle is doubled, the area of the new rectangle formed_______. (A) Remains the same (B) Is doubled (C) Becomes four times (D)Is halved EXERCISE 10.2. AREA 5

4. [AS1] The side of a square of area 256 cm2 is ________. (A) 12 cm (B) 16 cm (C)18 cm (D)20 cm 5. [AS1] The length of a rectangle of area 135 cm2 and breadth 9 cm is ________. (A) 11 cm (B) 12 cm (C)13 cm (D)15 cm Short Answer Type Questions 6 [AS1] In a rectangle, if l = 40 cm and its perimeter, P = 130 cm, find the breadth, b and the area, A of the rectangle. 7(i) [AS1] A rectangle has a perimeter of 20 cm. If the measures of sides are whole numbers in centimetres, ﬁnd the four possible pairs of values for the length and the width of the rectangle. (ii) [AS1] Convert: a) 2.45 m2 to cm2 b) 365 mm2 to cm2 8 [AS1] If the area of a square of side 16 cm is same as that of a rectangle of length 64 cm, what is the breadth of the rectangle? 9 [AS2] What will happen to the area of a rectangle if: (i) its length is doubled and breadth is trebled? (ii) its length and breadth are doubled? 10 [AS2] What will happen to the area of a square if its side is: (i) doubled? (ii) halved? 11(i) [AS4] Find the cost of distempering four walls of a room at the rate of Rs. 20 per sq. m, if each wall is square shaped of side 4 m. (ii) [AS4] A tile measures 10 cm × 10 cm. How many such tiles are required to cover a wall of dimensions 4 m × 2.5 m? 12 [AS4] Find the perimeter of a rectangular ﬁeld whose length is four times its width and which has an area 30976 sq. cm? EXERCISE 10.2. AREA 6

13 [AS4] It costs Rs. 1000 to fence a square ﬁeld at Rs. 2.50 per metre. Find (i) the length of the side of the square. (ii) the area of the square. 14(i) [AS4] A play ground measures 300 m by 170 m. Find the cost of planting grass on this at the rate of Rs. 80 per hectare. (ii) [AS4] A square piece of ground is 75 m long. Find the cost of erecting a fence around it at Rs. 4 per metre. Long Answer Type Questions 15 [AS1] (i) The area of a rectangle is 240 cm2 . If its length is 20 cm, ﬁnd its breadth. (ii) Find the perimeter of a rectangle whose area is 650 cm2 and breadth is 13 cm. 16 [AS1] Find the perimeter of a square whose area is 2500 m2 . 17 [AS2] What happens to the area of a rectangle when: (i) its breadth is doubled, the length remaining the same? (ii) its length and breadth both are doubled? 18 [AS2] What will happen to the area of a square if its side is trebled? 19 [AS4] The area of a square ﬁeld is 100 hectares. Find its side and perimeter. 20 [AS4] Four square ﬂower beds each of side 1.5 m are dug on a piece of land 8 m long and 5 m wide. What is the area of the remaining part of the land? 21 [AS4] The length of a rectangular plot of land is twice its breadth. If the perimeter of the plot is 300 m, ﬁnd its area. 22 [AS4] What is the area of a square shaped handkerchief of side 40 cm? 23 [AS4] The sides of a rectangular ground are 300 m and 120 m. Find its area in hectares. 24 [AS4] Find the area in hectares of a rectangular ﬁeld whose length is 240 m and breadth is 110 m. 25 [AS4] How many envelopes can be made out of a sheet of paper 125 cm by 85 cm, if each envelope requires a piece of paper of size 17 cm by 5 cm? EXERCISE 10.2. AREA 7

26 [AS4] A farmer has a rectangular ﬁeld which measures 350 m by 240 m. He hopes that by sowing a variety of wheat, the yield will be 25 quintals per hectare, and it will sell in the market atRs. 160 per quintal. What is his expected income? 27 [AS4] The length and breadth of a playground are 52 m 20 cm and 34 m 8 cm respectively. Find the cost of levelling it at Rs. 1.50 per square metre. 28 [AS4] Two plots of land have the same perimeter. One is a square of side 60 m, while the other is a rectangle whose breadth is 1.5 m. Which of the plots has greater area and by how much? 29 [AS4] A room is 12 m long and 8 m broad. It is surrounded by a verandah which is 2 m wide all over. Find the cost of ﬂooring the verandah with marble at Rs. 25 per square metre. 30 [AS4] A person walks at 4 km/h. How long will he take to go round a square ﬁeld 4 times, if its area is 900 m2 ? 31 [AS4] A rectangular court yard is 3 m 78 cm long and 5 m 25 cm broad. It is desired to pave it with square tiles of the same size. What is the largest size of the tile that can be used? Also ﬁnd the number of such tiles. 32 [AS5] (i) Find the area of the following ﬁgure by counting the squares. (ii) Find the area of the following ﬁgure by counting the squares. EXERCISE 10.2. AREA 8

CHAPTER 11 RATIO AND PROPORTION EXERCISE 11.1 COMPARING QUANTITIES WITH DIFFERENT UNITS 11.1.1 Key Concepts i. A type of comparison where we compare things or quantities by division is called ratio. ii. When we compare two quantities, we have to take care of the order of the quantities. iii. In general, the ratio of two quantities a and b is written as a : b and we read it as a is to b. iv. a : b b : a unless a = b. 11.1.2 Additional Questions Objective Questions 1. [AS3] In 13 : 17, 17 is called the . (A) Antecedent (B) Consequent (C)Middle term (D)Extreme 2. [AS1] If a = 2b, then a : b = . (A) 2 : 1 (C)3 : 4 (B) 1 : 2 (D)4 : 3 3. [AS1] The ratio of Rs. 2.50 and Rs. 3.75 is . (A) 3 : 2 (B) 4 : 5 (C)6 : 7 (D)2 : 3 EXERCISE 11.1. COMPARING QUANTITIES WITH DIFFERENT UNITS 9

4. [AS1] If 2x = 5y, then x : y is . (A) 5 : 2 (C)3 : 4 (B) 2 : 5 (D)5 : 9 5. [AS1] The ratio 384 : 480 in its simplest form is . (A) 3 : 5 (B) 5 : 4 (C)4 : 5 (D)2 : 5 Very Short Answer Type Questions 6. [AS3] Match the statements on ratio under column A with the appropriate answers given undercolumn B. Column A units. Column B i. A ratio has a. 2500 : 1 ii. The ratio of 2 cm to 10 mm b. order of quantities iii. The ratio of 5 g to 2 mg c. 2 : 1 iv. In a ratio is very important. d. “is to” ( : ) v. The symbol for ratio e. No Short Answer Type Questions 7 [AS4] John has 25 pens and Abhishek has 5 pencils. Find the ratio of the number of pens that John has to the number of pencils that Abhishek has. EXERCISE 11.1. COMPARING QUANTITIES WITH DIFFERENT UNITS 10

8 [AS5] In the ﬁgure, ﬁnd the ratio of number of white squares to the number of black squares and also write their number. Long Answer Type Questions 9 [AS4] Rani bought 5 kg of rice, Asha bought 4 kg of oil and Naresh bought 50 grams of cashew nuts. Find the ratio of rice to oil and cashew nuts. EXERCISE 11.1. COMPARING QUANTITIES WITH DIFFERENT UNITS 11

EXERCISE 11.2 RATIO IN DIFFERENT SITUATIONS 11.2.1 Key Concepts i. A ratio is said to be in the simplest form or in the lowest terms when it is written in terms of whole numbers with no common factors other than 1. For example, the simplest form of 35 : 40 is 7 : 8 (∵ 35 = 7×5 = 7 ) 40 8×5 8 11.2.2 Additional Questions . Objective Questions (B) 2 : 6 1. [AS1] A ratio equivalent to 2 : 3 is (A) 4 : 3 (C)6 : 9 (D)10 : 9 2. [AS1] The angles of a triangle are in the ratio 1 : 2 : 3. The measure of the largest angle is . (A) 30◦ (B) 60◦ (C) 90◦ (D) 120◦ 3. [AS4] The length and the breadth of a ﬁeld are in the ratio 5 : 3. If the breadth of the ﬁeld is 42 m, then its length is . (A) 50 m (B) 70 m (C)60 m (D)90 m 4. [AS4] Rahul has 3 marbles more than the number of marbles with Rajesh. If Rajesh has 12 marbles, then the ratio of marbles with Rajesh and Rahul is . (A) 4 : 5 (B) 5 : 4 (C)6 : 7 (D)7 : 6 EXERCISE 11.2. RATIO IN DIFFERENT SITUATIONS 12

5. [AS4] The ratio of boys and girls in a school is 12 : 5. If there are 840 girls in the school, then the number of boys is . (A) 1190 (B) 2016 (C) 2856 (D) 2142 Very Short Answer Type Questions (B) 3 : 5 6 [AS1] Choose the correct answer. (D)5 : 4 (i) The simplest form of 16 : 20 is _______. (B) 5 : 4 (A) 3 : 2 (D)3 : 4 (C)4 : 5 (B) 6 : 5 (ii) The simplest form of 18 : 24 is_______. (D)4 : 7 (A) 6 : 4 (C)4 : 4 (B) 3 : 4 (D)1 : 4 (iii) The simplest form of 12 : 28 is ________. (A) 7 : 5 (B) 7 : 11 (C)3 : 7 (D)4 : 9 (iv) The simplest form of 15 : 20 is ________. (A) 5 : 4 (C)2 : 4 (v) The simplest form of 21 : 33 is ______. (A) 3 : 5 (C)5 : 3 EXERCISE 11.2. RATIO IN DIFFERENT SITUATIONS 13

Short Answer Type Questions 7 [AS4] Let us assume that there are 56 students in a class of which 26 are girls. Find (i) The ratio of number of boys to number of girls. (ii) The ratio of number of boys to total number of students. (iii) The ratio of number of girls to total number of students. 8(i) [AS5] Finding the ratio among different quantities of chocolates: The given figure shows two boxes: One is Rani’s and the other is Priya’s. Observe the two boxes and write the ﬁrst quantity and second quantity. Also compare them using a statement. Find the ratio between these two boxes of chocolates. (ii) [AS4] Rajesh got 50 marks in Hindi, 75 marks in Maths and 80 marks in Social, each out of100 marks. Find the ratio of marks scored out of 100 by Rajesh in the three individual subjects. 9(i) [AS4] In a school, there are 22 boys and 33 girls. Find the ratio of number of boys to the number of girls and express in the simplest form. (ii) [AS4] In an exam, Raghu got 95 marks in Mathematics and 100 marks in Science each out of 100 marks. Find the ratio of marks in Mathematics to Science and express it in the simplest form. 10(i) [AS4] The ratio of male and female employees in a multinational company is 5 : 3. If there are115 male employees in the company, then ﬁnd the number of female employees. (ii) [AS4] Sunitha earned Rs. 40,000 and paid Rs. 5000 as income tax. Find the ratio of a) Income tax to income. b) Income to income tax. EXERCISE 11.2. RATIO IN DIFFERENT SITUATIONS 14

11(i) [AS5] In the given ﬁgure, ﬁnd the ratio of the number of: a) shaded parts to unshaded parts b) unshaded parts to shaded parts. (ii) [AS5] In the following ﬁgure, express the ratio of the number of shaded parts to those of the unshaded parts in the simplest form. 12 [AS5] Draw a four sided closed ﬁgure and divide it into the same number of equal parts horizontally and vertically. Shade the ﬁgure with any colour so that the ratio of the number of shaded parts to those of the unshaded parts is 1 : 7. EXERCISE 11.2. RATIO IN DIFFERENT SITUATIONS 15

EXERCISE 11.3 DIVISION OF A GIVEN QUANTITY IN A GIVEN RATIO 11.3.1 Key Concepts i. The ratio of two quantities ‘a’ and ‘b’ can be given in any one of the following ways: a. Symbolic form: a : b b. Fractional form: a b c. Verbal form: ‘a’ is to ‘b’ 11.3.2 Additional Questions Objective Questions 1. [AS1] If A, B and C divide Rs.1200 among themselves in the ratio 2 : 3 : 5, then B’s share is . (A) Rs. 240 (B) Rs. 300 (C)Rs. 350 (D)Rs. 360 2. [AS4] A bag of 88 marbles is shared between Raghu and Harish in the ratio 3 : 5. The number of marbles that Harish receives is _______. (A) 33 (B) 35 (C) 55 (D) 65 3. [AS4] 65 chocolates are divided between Manisha and Shravya in the ratio 5 : 8. Then the number of chocolates that Manisha gets is . (A) 25 (B) 40 (C) 38 (D) 52 EXERCISE 11.3. DIVISION OF A GIVEN QUANTITY IN A GIVEN RATIO 16

4. [AS4] When some amount is divided between Nishanth and Suresh in the ratio 4 : 5, Nishanth got Rs. 8000. Then Suresh’s share is . (A) Rs. 18,000 (B) Rs. 10,000 (C)Rs. 15,000 (D)Rs. 13,000 5. [AS1] Two numbers are in the ratio 5 : 6. If the sum of the numbers is 198, then the two numbers are . (A) 100, 98 (B) 80, 118 (C)120, 78 (D)90, 108 Short Answer Type Questions 6(i) [AS4] Rahul, Ravi and Sunil divide Rs. 24,000 among themselves in the ratio 1 : 3 : 4. Find the share of each. (ii) [AS4] A box of 25 pencils is shared between Akshay and Aditya in the ratio 2 : 3. a) How many pencils does Aditya receive? b) How many pencils does Akshay receive? EXERCISE 11.3. DIVISION OF A GIVEN QUANTITY IN A GIVEN RATIO 17

EXERCISE 11.4 PROPORTION AND UNITARY METHOD 11.4.1 Key Concepts i. Equalities of ratios is called proportion. In general, if the ratio of a and b is equal to the ratio of c and d, we say that they are in proportion. This is represented as a : b :: c : d. 11.4.2 Additional Questions Objective Questions 1. [AS1] If 4, a, a and 36 are in proportion, then a = . (A) 16 (B) 18 (C) 20 (D) 12 2. [AS3] If a, b, c and d are in proportion, then . (A) ab = cd (B) ac = bd (C)ad = bc (D)None of these 3. [AS3] If 14, 36, x and 72 are in proportion, then the value of x is . (A) 16 (B) 28 (C) 18 (D)None of these 4. [AS4] If the cost of 5 pens is Rs. 30, then the cost of one dozen pens is . (A) Rs. 60 (B) Rs. 120 (C)Rs. 72 (D)Rs. 140 EXERCISE 11.4. PROPORTION AND UNITARY METHOD 18

5. [AS1] The ﬁrst, second and fourth terms of a proportion are 16, 24 and 54 respectively. The third term is . (A) 32 (B) 48 (C) 28 (D) 36 Very Short Answer Type Questions 6. [AS1] Match the ratios in column A with their corresponding terms in column B. Column A Column B i. 2 : 3 : : 16 : a. Product of means ii. The cost of 6 erasers when the b. Proportion cost of 5 erasers is Rs. 20 c. 10 iii. Equality of two ratios iv. In a proportion, product of extremes = d. Rs. 24 v. 5 : 7 : : : 14 e. 24 Short Answer Type Questions 7(i) [AS2] Mahesh and Sai added their money and bought 20 eggs. Mahesh contributed Rs.12 and Sai Rs.18. They wanted to distribute the eggs between them. Sai said 10 eggs for each and Mahesh said 12 eggs for Sai and 8 eggs for him. Who is correct? Justify your answer. (ii) [AS4] If three oranges cost is Rs. 30, how much would 5 oranges cost? Long Answer Type Questions 8 [AS4] A farmer has sheep and cows in the ratio of 7 : 4. (i) How many sheep does the farmer have, if he has 240 cows? (ii) Find the ratio of number of sheep to the total number of animals the farmer has. (iii) Find the ratio of the total number of animals with the farmer to the number of cows with him. EXERCISE 11.4. PROPORTION AND UNITARY METHOD 19

(iv) How many animals did the farmer have in all? (v) Find the ratio between the number of cows and the number of sheep that the farmer has. 9 [AS4] The cost of 7 boxes of apples is Rs. 280. What is the cost of 4 boxes of apples? 10 [AS4] A box of fruits contains apples and oranges. For every 2 apples, there are 6 oranges. Complete the table based on the given information. Apples 24 6 Oranges Total no. of fruits 6 12 24 24 40 (i) What is the ratio of the number of apples to that of oranges? (ii) If you have 8 apples, how many oranges will you have? (iii) If there are 32 fruits in a medium-sized box, how many will be apples? (iv) In the super fat box, there are 40 fruits. How many will be oranges? (v) In the box, there are 12 apples. How many fruits are there in the box? 11 [AS4] (i) A train journey of 75 km costs Rs. 215. How much will a journey of 120 km cost? (ii) If the sales tax on a purchase worth Rs 60 is Rs. 4.20, what will be the sales tax on the purchase worth Rs. 150? EXERCISE 11.4. PROPORTION AND UNITARY METHOD 20

CHAPTER 12 SYMMETRY EXERCISE 12.1 LINE SYMMETRY 12.1.1 Key Concepts i. Any line along which we can fold a ﬁgure so that its two parts coincide exactly is called aline of symmetry or axis of symmetry. It can be horizontal, vertical or diagonal. ii. The dotted line in the given figure is a line of symmetry. The alphabet ‘A’ has vertical symmetry. The letters H, I, M, O, T, U, V, W, X and Y have vertical line of symmetry. iii. The alphabet ‘B’ has horizontal symmetry. The letters B, C, D, E, H, I, K, O and X have horizontal line of symmetry. iv. The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry. v. The line of symmetry is closely related to mirror reﬂection. When dealing with mirror reﬂection we have to take into account the left and right changes in orientation. vi. A ﬁgure may have no line of symmetry, only one line of symmetry, two lines of symmetry or multiple lines of symmetry. Here are some examples in the given table. EXERCISE 12.1. LINE SYMMETRY 21

S.No. Number of lines of symmetry Example 1. 0 A scalene triangle 2. 1 An isosceles triangle 3. 2 A rectangle 4. 3 An equilateral triangle 5. Infinitely many A circle 12.1.2 Additional Questions Objective Questions 1. [AS3] Any line along which we can fold a ﬁgure so that its two parts coincide exactly is called its . (A) Median (B) Altitude (C)Perpendicular bisector (D) Line of symmetry 2. [AS3] A line of symmetry is also known as . (A) An axis of symmetry (B) A point of symmetry (C)A perpendicular bisector (D)None of these EXERCISE 12.1. LINE SYMMETRY 22

3. [AS5] The number of lines of symmetry for the given ﬁgure is . (A) 1 (B) 2 (C) 0 (D) 3 4. [AS3] The number of lines of symmetry for the letter A is . (A) 1 (B) 2 (C) 3 (D) 4 5. [AS3] The number of lines of symmetry for the letter O is . (A) 1 (B) 2 (C) 3 (D)Inﬁnitely many Very Short Answer Type Questions 6. [AS5] Match the letter with its mirror image. Column A Column B i. a. ii. b. iii. c. iv. d. v. e. EXERCISE 12.1. LINE SYMMETRY 23

7 [AS5] Answer the following questions in one sentence. 24 Identify the line of symmetry for the following figures. (i) (ii) (iii) (iv) (v) EXERCISE 12.1. LINE SYMMETRY

8 [AS5] Answer the following questions in one sentence. Complete the ﬁgures such that the dotted line is the line of symmetry. (i) (ii) (iii) (iv) (v) EXERCISE 12.1. LINE SYMMETRY 25

EXERCISE 12.2 MULTIPLE LINES OF SYMMETRY 12.2.1 Key Concepts i. Symmetry has plenty of applications in everyday life as in art, architecture, textile technology, design creations, geometrical reasoning, kolams, rangolis etc. ii. Observe the lines of symmetry in the following ﬁgures. In each ﬁgure, the dotted line represents the line of symmetry. All the given ﬁgures have vertical line of symmetry. iii. There are ﬁgures that have horizontal line or slant line of symmetry. iv. Some figures have multiple lines of symmetry. 12.2.2 Additional Questions Objective Questions 1. [AS5] The number of lines of symmetry of a kite is . (A) 1 (B) 2 (C) 3 (D) 4 2. [AS5] The number of lines of symmetry of the letter Z is . (A) 0 (B) 1 (C) 2 (D) 3 3. [AS5] The number of lines of symmetry of a regular hexagon is . (A) 2 (B) 3 (C) 5 (D) 6 EXERCISE 12.2. MULTIPLE LINES OF SYMMETRY 26

4. [AS3] A rhombus is symmetrical about . (A) Each of its diagonals (B) The line joining the midpoints of its opposite sides (C) The perpendicular bisectors of each of its sides (D)None of these 5. [AS3] An equilateral triangle is symmetrical about each of its . (A) Altitudes (B) Medians (C)Angle bisectors (D)All of these EXERCISE 12.2. MULTIPLE LINES OF SYMMETRY 27

Very Short Answer Type Questions 6. [AS5] Match the ﬁgures in column A with the ones with lines of symmetry in column B. Column A Column B i. a. ii. b. iii. c. iv. d. v. e. EXERCISE 12.2. MULTIPLE LINES OF SYMMETRY 28

7 [AS5] Choose the correct answer. (i) The number of lines of symmetry of a rectangle is _______. (A) 4 (B) 2 (C) 3 (D) 1 (ii) The number of lines of symmetry of a square is _______. (A) 1 (B) 2 (C) 3 (D) 4 (iii) The number of lines of symmetry of a parallelogram is _________. (A) 0 (B) 2 (C) 3 (D)Inﬁnitely many (iv) The number of lines of symmetry of a rhombus is _______. (A) 1 (B) 2 (C) 3 (D) 4 (v) The number of lines of symmetry of a regular octagon is _______. (A) 6 (B) 2 (C) 8 (D) 4 EXERCISE 12.2. MULTIPLE LINES OF SYMMETRY 29

CHAPTER 13 PRACTICAL GEOMETRY EXERCISE 13.1 A LINE SEGMENT 13.1.1 Key Concepts i. The geometrical instruments used to construct shapes: a. A graduated ruler b. The compass c. The divider d. Set–squares e. The protractor 13.1.2 Additional Questions Objective Questions 1. [AS3] A line segment has a deﬁnite . (A) Length (B) Breadth (C)Height (D) None of these 2. [AS3] The distance between the points A and B is called the of AB. (A) Breadth (B) Length (C)Height (D)None of these 3. [AS1] If AB = 6.7 cm and C is the mid–point of AB, then AC = . (A) 3.5 cm (B) 3.25 cm (C)3.35 cm (D)3.15 cm EXERCISE 13.1. A LINE SEGMENT 30

4. [AS3] Line segment AB is line segment BA. (A) Equal to (B) Not equal to (C)Perpendicular to (D)None of these 5. [AS3] The length of a line segment is the distance between its two end points. (A) Maximum (B) Shortest (C)Congruent (D)None of these Short Answer Type Questions 6(i) [AS1] Given that AB = 12 cm, AD = 18 cm and AE = 20 cm, ﬁnd BC, CD and DE. The distance between BC and CD are equal as shown in the ﬁgure. (ii) [AS1] Given that PQ = 6 cm, mark a point R on PQ. Find the length between PR and QR and assume R is about 5 cm away from P on the line. 7(i) [AS5] If AB = 7.5 cm and CD = 2.5 cm, construct a line segment whose length is equal to a) AB − CD b) 2AB (ii) [AS5] Draw a line segment CD. Produce it to CE such that CE = 3CD. 8 [AS5] Draw a line segment AC of length 5 cm. Mark a point B on AC and ﬁnd the length between AB and BC. Assume that ‘B’ is the midpoint of AC. 9(i) [AS5] By using ruler, construct a line segment of length 5.9 cm. (ii) [AS5] Construct a line segment RS such that it is three times that of PQ which is 2 cm. 10 [AS5] Construct a line segment CD of length 12 cm. Mark a point ‘O’ on it at a distance of 6 cm from C. Measure CO and OD. What do you observe? 11(i) [AS5] Construct a line segment of length 7.2 cm using compass. (ii) [AS5] PQ = 2.4 cm. Construct RS using compass such that the length of RS is four times that of PQ. EXERCISE 13.1. A LINE SEGMENT 31

EXERCISE 13.2 CONSTRUCTION OF A CIRCLE 13.2.1 Key Concepts i. Every point on the boundary of a circle is equidistant from its centre. 13.2.2 Additional Questions Objective Questions 1. [AS3] The geometrical instrument we use to construct a circle is________. (A) A ruler (B) A compass (C) A divider (D)A protractor 2. [AS3] A set of points in a plane which are at the same distance from a ﬁxed point is called a . (A) Rectangle (B) Rhombus (C)Square (D)Circle 3. [AS3] The distance between the centre of a circle and any point on it is called its . (A) Radius (B) Diameter (C) Chord (D) Tangent 4. [AS3] A circle can have radii. (A) Two (B) Three (C) Five (D)Inﬁnitely many EXERCISE 13.2. CONSTRUCTION OF A CIRCLE 32

5. [AS3] Two circles having the same radii are called circles. (A) Concentric (B) Congruent (C)Similar (D)None of these Very Short Answer Type Questions 6 [AS3] Answer the following questions in one sentence. What is meant by concentric circles ? 7 [AS5] Answer the following questions in one sentence. (i) Construct a pair of concentric circles of radii 3 cm and 5 cm. (ii) Construct a set of concentric circles of radii 3 cm, 4 cm and 5 cm. (iii) Construct a pair of concentric circles of radii 2.5 cm and 5 cm. (iv) Construct a set of concentric circles of radii 4 cm, 5 cm and 6 cm. Short Answer Type Questions 8(i) [AS5] Construct a circle with centre ‘O’ and radius 2.5 cm. (ii) [AS5] Construct a circle with centre ‘P’ and radius 3 cm. 9(i) [AS5] Construct three circles in such a way that the circles intersect at four points. (ii) [AS5] Construct 5 circles in such a way that the circles intersect at 8 points. EXERCISE 13.2. CONSTRUCTION OF A CIRCLE 33

EXERCISE 13.3 PERPENDICULARS 13.3.1 Key Concepts i. Two lines are said to be perpendicular if they intersect such that the angles formed between them are right angles. 13.3.2 Additional Questions Objective Questions 1. [AS3] If two lines intersect at right angles, then they are called . (A) Parallel lines (B) Perpendicular lines (C)Transversal lines (D)None of these 2. [AS3] The symbol for ‘is perpendicular to’ is . (A) = (B) ⊥ (C) (D)None of these 3. [AS3] A divides a line segment into two equal parts. (A) Bisector (B)Perpendicular (C)Divider (D)None of these 4. [AS3] The line which is perpendicular to the given line segment and also passes through its mid–point is called its . (A) Perpendicular bisector (B) Angular bisector (C)Median (D)None of these EXERCISE 13.3. PERPENDICULARS 34

5. [AS3] To draw the perpendicular bisector, we use . (A) Ruler and divider (B) Ruler and compass (C) Ruler only (D)None of these Very Short Answer Type Questions 6. [AS2] Estimate if one of the given lines is perpendicular to the other line or not. (i) (ii) (iii) (iv) (v) EXERCISE 13.3. PERPENDICULARS 35

Long Answer Type Questions 7 [AS5] Draw a line segment of 15 cm and draw a perpendicular bisector of the given line segment. 8 [AS5] Draw a line segment AB of any length and ﬁnd its mid point. 9 [AS4] Find the midpoint of CD which is the edge of a science text book. 10 [AS5] Using a ruler and compass, draw AB = 7.8 cm. Find its mid point. EXERCISE 13.3. PERPENDICULARS 36

EXERCISE 13.4 CONSTRUCTION OF ANGLES USING PROTRACTOR 13.4.1 Key Concepts Using a ruler and a compass, the following can be constructed: i. A circle, when the length of its radius is known e)120◦ ii. A line segment, if its length is given iii. A copy of a line segment iv. A perpendicular to a line through a point a. On the line b. Not on the line v. The perpendicular bisector of a line segment of a given length vi. An angle of a given measure vii. A copy of an angle viii. The bisector of a given angle ix. Some angles of special measures as: a)30◦ b)60◦ c)45◦ d)90◦ 13.4.2 Additional Questions Objective Questions 1. [AS3] The mathematical instrument which is used to construct angles is . (A) Ruler (B) Divider (C)Protractor (D)None of these 2. [AS3] An angle is measured in . (A) Degrees (B) Metres (C) Litres (D)Kilograms EXERCISE 13.4. CONSTRUCTION OF ANGLES USING PROTRACTOR 37

3. [AS3] The ray which divides the given angle into two equal measures is called . 38 (A) An angle bisector (B) A perpendicular bisector (C)A median (D)The centroid 4. [AS1] ∠AOB = 90◦ and OD is the bisector of ∠AOB. Then ∠AOD = . (A) 90◦ (B) 60◦ (C) 75◦ (D) 45◦ 5. [AS3] An angle whose measure is 90◦ is called . (A) An acute angle (B) An obtuse angle (C)A right angle (D)A reﬂex angle Short Answer Type Questions 6(i) [AS5] Construct ∠PQR = 85◦. Then construct the bisector of ∠PQR. (ii) [AS5] Construct ∠ABC = 90◦. Then construct the bisector of ∠ABC. Long Answer Type Questions 7 [AS5] Construct the following angles using a protractor: (i) ∠ABC = 35◦ (ii) ∠DEF = 40◦ (iii) ∠GHI = 110◦ (iv) ∠JKL = 145◦ (v) ∠MNO = 150◦ EXERCISE 13.4. CONSTRUCTION OF ANGLES USING PROTRACTOR

EXERCISE 13.5 CONSTRUCTING ANGLES OF SPECIAL MEASURES 13.5.1 Key Concepts i. Some special angles can be drawn without using a protractor. ii. We can bisect any angle using a compass. 13.5.2 Additional Questions Objective Questions 1. [AS3] Some angles of special measures such as 90◦, 45◦, 60◦, 30◦, 120◦ and 135◦ can be drawn using . (A) A ruler and divider (B) A ruler and a compass (C)Set squares (D)None of these 2. [AS3] An angle whose measure is 180◦is called a/an . (A) Right angle (B) Straight angle (C)Acute angle (D)Obtuse angle 3. [AS3] An angle whose measure is 360◦ is called a . (A)Right angle (B) Straight angle (C)Complete angle (D)Reﬂex angle 4. [AS3] An angle whose measure is less than 90◦ and greater than zero degrees is called . (A) A right angle (B) An acute angle (C)An obtuse angle (D)A straight angle EXERCISE 13.5. CONSTRUCTING ANGLES OF SPECIAL MEASURES 39

5. [AS3] When two rays meet at a point, they form a / an . (A) Line segment (B) Angle (C) Line (D)None of these Short Answer Type Questions 6(i) [AS5] With the help of a compass and a ruler construct an angle of ∠ABC = 60◦. (ii) [AS5] With the help of a compass and a ruler, construct an angle of 120◦. 7(i) [AS5] Using a compass and a ruler, construct an angle of 45◦. (ii) [AS5] Using a compass and a ruler, construct the following angles: a) An angle of 75◦ b) An angle of 90◦ c) An angle of 150◦ 8(i) [AS5] Construct ∠PQR = 30◦ and draw another angle ∠ABC = ∠PQR using compass. (ii) [AS5] Construct ∠ABC = 60◦ and then draw another angle ∠XYZ = ∠ABC without using a protractor. EXERCISE 13.5. CONSTRUCTING ANGLES OF SPECIAL MEASURES 40

CHAPTER 14 UNDERSTANDING 3D AND 2D SHAPES EXERCISE 14.1 3–D SHAPES 14.1.1 Key Concepts i. Certain objects have only two dimensions, i.e., length and breadth. They are called two dimensional or 2D objects. Example: Timetable card, Exam paper, Brown sheet etc. ii. Some objects have three dimensions, i.e., length, breadth and height (thickness). They are called three dimensional or 3D objects. Example: Textbook, Eraser, Geometry box, Matchbox etc. iii. Objects with length, breadth and height / thickness are called solids. A cuboid has 6 faces, 8 vertices and 12 edges. iv. A solid formed by the enclosure of six square plane surfaces is called a cube. A cube has 6 faces, 8 vertices and 12 edges. EXERCISE 14.1. 3–D SHAPES 41

v. Cylinder: Tins, oil drums, wooden logs etc., are in the shape of a cylinder. The base and top of a cylinder are circular in shape. The remaining surface is called its curved surface. It has no straight edges. vi. Cone: Ice–cream cones, joker’s caps are in the shape of cone. The base of a cone is a circle. In the given figure of a cone, ‘A’ is called its 'vertex'. AC is called its 'slant height' denoted by ‘s’ and AO is called its 'vertical height' denoted by ‘h’. OC is its radius denoted by ‘r’. It has no straight edges. vii. Sphere: A ball, a laddoo etc. are in the shape of a sphere. The surface of a sphere is called its curved surface. In the given figure of a sphere, ‘O’ is called its “centre”. AO is called its radius. It has no straight edges. Football, Volley ball, Table tennis ball and Lawn tennis ball are all spheres. EXERCISE 14.1. 3–D SHAPES 42

viii. Prism –Triangular prism: The base of the prism in the given figure is a triangle. So, it is called a triangular prism. It has 5 surfaces and 6 vertices. ix. Square pyramid: In a square pyramid, the base is a square. Its lateral surfaces are triangular in shape, such that all the four surfaces meet at a point. It has ﬁve surfaces. x. Triangular pyramid: If a pyramid has a triangular base, then it is called a “Triangular pyramid”. It has three triangular lateral surfaces and one triangular base. Altogether it has four triangular faces. ‘A’ is called “apex or vertex”. AE is called its height. EXERCISE 14.1. 3–D SHAPES 43

14.1.2 Additional Questions Objective Questions 1. [AS3] A cuboid has edges. (A) 2 (C) 4 (B) 3 (D) 12 2. [AS3] The number of lateral faces of a cuboid is . (A) 4 (B) 5 (C) 6 (D) 7 3. [AS4] A brick is an example of a . (A) Cube (B) Cuboid (C)Cylinder (D)Cone 4. [AS3] The number of edges of a triangular pyramid is . (A) 3 (B) 4 (C) 6 (D) 8 5. [AS3] A is a solid shape with a base, a vertex and triangular faces. (A) Pyramid (B) Prism (C)Cone (D)Cylinder EXERCISE 14.1. 3–D SHAPES 44

Very Short Answer Type Questions 6. [AS5] Match the 3D ﬁgures in column A with their names in column B. Column A Column B i. a. Cylinder ii. b. Cuboid iii. c. Prism iv. d. Pyramid v. e. Cube EXERCISE 14.1. 3–D SHAPES 45

7 [AS5] Answer the following questions in one sentence. Count the number of faces, edges and vertices for the 3–D objects given. (i) (ii) (iii) (iv) (v) EXERCISE 14.1. 3–D SHAPES 46

8 [AS5] Answer the following questions in one sentence. Count the number of curved surfaces, plane surfaces and vertices of the solids given. (i) (ii) (iii) (iv) (v) Short Answer Type Questions 9(i) [AS2] A die is an example of a cube. Write the number of vertices, faces and edges of a die. Is it the same as in the case of a cuboid? (ii) [AS2] A match box is an example of a cuboid. Write the number of vertices, faces and edges of the match box. Is it the same as in the case of a cube? 10(i) [AS5] Draw a cone and name its radius, vertical height and slant height. (ii) [AS5] Draw a cylinder and name its radius and height. EXERCISE 14.1. 3–D SHAPES 47

EXERCISE 14.2 POLYGONS 14.2.1 Key Concepts i. Polygon: A polygon is a closed ﬁgure made up of line segments. ii. Regular polygon: A polygon with all equal sides and all equal angles is called a regular polygon. Equilateral triangles and squares are examples of regular polygons. 14.2.2 Additional Questions Objective Questions 1. [AS3] A closed ﬁgure formed with line segments is called a . (A) Circle (B) Polygon (C)Cone (D)None of these 2. [AS3] A polygon with all sides equal and all angles equal is called a . (A) Quadrilateral (B) Cube (C)Triangle (D)Regular polygon EXERCISE 14.2. POLYGONS 48

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