GEOMETRY 295 CLASSWORK EXAMPLES Example : 1 Copy the given line segment PQ below and draw its image by a translation 6 cm to the right. Solution: (i) Draw a horizonal lines P and Q respectively. (ii) Take a point P' and P at 6 cm to the right. P Q Q' q p P' (iii) Take points Q' on q at distance 6 cm to the right. (iv) Joint P' and Q', we get the image P'Q' of PQ. We wrote PQ →T P'Q'. We read it as PQ is translated to P'Q'. Example : 2 A (2, 3), B (5, 4) and C (3, –2) are the vertices of ∆ABC. Find the image of ∆ABC under the translation –2 3 T . Represent the object and its image by graph. Solution: Here, the vertices of DABC are A (2, 3), B (5, 4) and C (3, –2). Translating the vertices of DABC by –2 3 T , we get A(2, 3) –2 3 T A'(2 – 2, 3 + 3) = A'(0, 6) B(5, 4) –2 3 T B' (5 – 2, 4 + 3) = B'(3, 7) C(3, –2) –2 3 T C' (3 – 2, – 2 + 3) = C'(1, 1) The object and image so obtained are graphed alongside. P Q C C' B' A' B A Y' X' X Y
296 The Leading Maths - 8 EXERCISE 17.2 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is translation? Define with an example. (b) Write three examples of translation. (c) When a point A(r, s) is translated under the vector p q , what is its image? (d) Write the image of a point M(– 1, 2) under translation vector 2 –2 . 2. Circle ( ) the correct answers. (a) The image of P (r, y) under translation vector a b is .............. i. P'(y, x) ii. p'(a – x, b – y) iii. (x + a, y + b) iv. (x – a, y – b) (b) What is the image point of M (o, u) when it is translating by 1 3 ? i. (1, 3) ii. (2, 3) iii. (1, 3) iv. (3, 1) (c) Which is the image of the point (2, 3) when translating by –2 3 ? i. (4, 6) ii. (0, 3) iii. (0, 6) iv. (4, 3) (d) When the point (–3, –1) is translated by the vector –1 –2 , which is its image point? i. (– 4, – 3) ii. (– 2, – 2) iii. (4, 3) iv. (– 2, – 3) 3. T = a is a translation. Find the image of the following objects after translating by T = a . (a) A B C a (b) D F E a (c) G H I a (d) a (e) (f) a a
GEOMETRY 297 4. Observe the figures carefully and complete the following table for the translation (a) and (b). (a) (b) (a) (b) Pre-image Image Pre-image Image A(2, 4) → A'(.............) A(5, 5) → A'(.............) B(.............) → B'(.............) B(.............) → B'(.............) C(.............) → C'(.............) C(.............) → C'(.............) D(.............) → D'(.............) Write the translation component in the form a b T for the translations (a) and (b) 5. Repeat the question no. 4 for the following translation: X' X Y Y' O A B C B' A' C' X' X Y Y' O B B' C C' D D' A A' 6. (a) A(2, 1), B(4, 0) and C(3, 5) are the vertices of ∆ABC. Find its image by stating coordinates and by graphing under a translation T 4 0 . (b) Find the coordinate of the verices of the image of a quadrilateral having vertices P(1, 0), Q(5, 3), R(4, 5) and S(2, 4) under the translation by the vector – 2 0 . 7. Find the image of ∆ABC with vertices A(-2, -3), B(0, 3) and C(3, 1) under the following translations. Also show them in the graph. (a) 0 4 T (b) 0 –5 X' X Y' Y' O A B C B' A' C' X' Y' Y' X O B' B C' C D' D A' A
298 The Leading Maths - 8 8. A(–1, 2), B(–2, –1), C(2, –2) and D(1, 1) are the vertices of a quadrilateral ABCD. Find by stating coordinates and drawing graph the image of quadrilateral ABCD by the following translations. (a) –2 4 T (b) 4 –2 T 9. T(–1, 1), R(–2, –1), A(3, –1) and P(1, 1) are the vertices of a trapezium TRAP. Find by stating coordinates and by graphing the image of the trapezium TRAP under the following translations vectors. (a) 2 –1 T (b) –4 –5 T 10. K(3, 1), I(1, –1), T(3, –3) and E(2, –1) are the vertices of a kite. Find the image of the kite KITE under the following translation by graphing the images. (a) 4 0 (b) 0 –4 ANSWERS 5. (a) A(1, 1), B(– 1, – 1), C(2, – 2); A'(– 3, – 1) B'(– 5, – 3), C'(– 1, – 4) (b) A(2, 3), B(0, – 1), C(4, 1), D(2, 1); A'(– 1, 4), B'(– 3, 0), C'(1, 2), D'(– 1, 2) 6. (a) A'(6, 1), B'(8, 1), C'(7, 5) (b) P'(– 1, 0), Q'(3, 3), R'(2, 5), S'(0, 4) 7. (a) A'(– 2, 1), B'(0, 7), C'(3, 5), (b) A'(– 2, – 8), B'(0, – 2), C'(3, – 4) 8. (a) A'(– 3, 6), B'(– 4, 3), C'(0, 2), D'(– 1, 5) (b) A'(3, 0), B'(2, – 3), C'(6, – 4), D'(5, – 1) 9. (a) T'(1, 0), R'(0, – 2), A'(5, – 2), P'(3, 0) (b) T'(– 5, – 4), R'(– 6, – 6), A'(– 1, – 6), P'(– 3, – 4) 10. (a) K'(7, 1), I'(5, – 1), T'(7, – 3), E'(6, – 1) (b) K'(3, – 3), I'(1, – 5), T'(3, – 7), E'(2, – 5) Project Work 17.2 Draw a triangle on a graph. Translate it by a vector 2 3 then translate its image by another vector –3 1 on the same graph paper. Write the coordinates of the object triangle and its both images. Notice the coordinates of the object triangle and final image triangle. Write your conclusion.
GEOMETRY 299 At the end of this topic, the student will be able to: ¾ find the coordinate of the image of the given objects under the given rotation. Learning Objectives 17.3 Rotation I. Introduction When a given figure is turned at a given angle about a fixed point, the movement is called a rotation. If the ∆ABC is turn 90° in anti-clockwise direction, the position of the point A will be A' and B will be at B', we denote AB R° (+ 90°) A'B' A rotation is a transformation that maps every point in the plane to its image by rotating the plane around a fixed point. The fixed point is called the center of the rotation and the turning amount of rotation is called the angle of rotation. When you rotate AB through 90° in anti-clockwise direction about A, the new position of B will be at B'. B RA (+ 90°) B' For rotation in with centre A though 90° in clockwise direction about A, the new position of B will be B'. B RA (– 90°) B' II. Rotation Using Coordinates a. Positive Quarter Turn About Origin 'O' In the figure, ∆A'B'C' is the image of ∆ABC after a rotation of 90° in anticlockwise direction about O. This is called positive quarter turn. Complete the following table from the given graph. A B B' 90° A B B' 90° Y Y' X' B A' B' O X A
300 The Leading Maths - 8 Pre-image Rotation Image A(2, 4) Ro (+ 90°) A'( , ) B( , ) Ro (+ 90°) B'( , ) C( , ) Ro (+ 90°) C'( , ) ∴P (a, b) Ro (+ 90°) P'( , ) In general, (x, y) RO(+ 90°) (–y, x) b. Negative Quarter Turn About origin 'O' In the figure, ∆A'B'C' is the image of ∆ABC after a negative quarter turn about origin O i.e. Ro (– 90°). Complete the following table. Pre-image Rotation Image A( , ) R° (– 90°) A'( , ) B( , ) R° (– 90°) B'( , ) C( , ) R° (– 90°) C'( , ) P( , ) R° (– 90°) P'( , ) In general, (x, y) R° (– 90°) (y, – x) c. Half Turn About Origin 'O' In the figure, DP'Q'R' is the image of DPQR after half turn about origin O. Read the coordinates of the object and image points and complete the following table. Pre-image Rotation Image P(1, 5) R° (+ 180°) P'( , ) Q( , ) R° (+ 180°) Q'( , ) R( , ) R° (+ 180°) R'( , ) A(a, b) R° (+ 180°) A'( , ) In general, (x, y) R° (+ 180°) (–x, –y) X Y O A' A B C' B' C –90° X Y Y' X' O 180° Q(2, 2) P(1, 5) P'(–1, –5) R'(–6, –3) Q'(–2, –2) R(6, 3) X Y Y' X' O A A' C C' B' B
GEOMETRY 301 CLASSWORK EXAMPLES Example : 1 Draw the image of ∆ABC for RO (– 90°). Solution: First find the image of A; 1. Draw OA. 2. Draw an angle of 90° at O. 3. Locate A' such that OA' = OA. A' is the image of A. Then find the image of B and C as B' and C' respectively and join A', B' and C' to form ∆A'B'C'. i.e. ∆ABC R° (– 90°) ∆A'B'C' Example : 2 A(2, 1), B(4, –1) and C(3, 4) are vertices ∆ABC. Find ∆ABC RO+ 90° ∆A'B'C'. Solution: Rotating the vertices of ∆ABC under the rotation through +90° about the origin O, we have (x, y) R° (+ 90°) (-y, x) ∴ A(2, 1) R° (– 90°) A'(–1, 2) B(4, –1) R° (– 90°) B'(1, 4) C(3, 4) R° (– 90°) C'(–4, 3) The object and its image are graphed in the figure. Example : 3 P(2, 5), Q(4, 8), R(3, –2) are the vertices of ∆PQR. Find ∆PQR R° (– 90°) ∆P'Q'R' and graph the object and its image. Solution: Reflecting the vertices ∆PQR under the rotation through – 90° about the origin O, X' X Y Y' O A C A' B B' C' X Y' X' Y O C(3, 4) A' C' B' A(2, 1) B(4, –1) 90+ X Y Y' X' O P Q R' P' Q' R – 90°
302 The Leading Maths - 8 we have (x, y) R° (– 90°) (y, -x) P(2, 5) R° (– 90°) P'(5, –2) Q(4, 8) R° (– 90°) Q'(8, –4) R(3, –1) R° (– 90°) R'(–1, –3) The object and image so obtained are graphed in the figure. Example : 4 Find the image by stating coordinate and by graphics for the ∆ABC with vertices A(1, –2), B(5, 0) and C(0, 4) after half turn about origin. Solution: Reflecting the vertices of ∆ABC under half turn about origin, we have (x, y) R° (180°) (-x, -y) ∴ A(1, –2) R° (180°) A'(–1, 2) B(5, 0) R° (180°) B'(–5, 0) C(0, 4) R° (180°) C'(0, –4) The object and its image are graphed in the figure. EXERCISE 17.3 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is rotation? Write two examples of rotation used in daily life. (b) What is the image of a point (x, y) under the rotation through – 90° about the origin? (c) Write the image of a point P(– 3, 3) under the rotation through + 90° about O. (d) Write the image of a point M(– 1, 2) under translation vector 2 –2 . X Y Y' X' B(5, 0) A(1, –2) C(0, 4) A'(–1, 2) C'(0, –4) B'(–5, 0) O
GEOMETRY 303 2. Circle ( ) the correct answers. (a) Which is the correct image of a point (x, y) after rotating through + 90° about the origin O? i. (y, x) ii. (y, – x) iii. (– y, x) iv. ( – y, x) (b) What is the image of a point (p, q) under the rotation through 180° about O? i. (q, p) ii. (– q, – p) iii. (–p, –q) iv. (– q, p) (c) Which is the image of the point (4, – 3) under the rotation through – 90° about the centre as origin? i. (3, 4) ii. (– 3, – 4) iii. (–3, 4) iv. (3, –4) (d) Which is the image of the point (– 3, – 2) after rotating it through positive quarter turn around the origin? i. (2, – 3) ii. (2, 3) iii. (– 2, – 3) iv. (3, 2) 3. Find the image of the following shapes after rotating about O at the given angle. (a) (+ 90°) A B (b) A B (+ 90°) O (c) A B (– 90°) O (d) B O A (180°) (e) (f) A O B (180°) C B A C O (+ 90°)
304 The Leading Maths - 8 4. Find the image of the following shapes after rotating them about O at the given angle. (a) (+ 90°) P Q (b) O(+ 90°) (c) O(+90° +) (d) O (180°) (e) O (– 90°) (f) 5. Find the image of the following triangle after rotation about O at the given angle. (a) (d) (b) (e) (c) (f) A A A A A O O O O O O B B B B B (+90°) (+90°) (–90°) (180°) (–90°) 180° P 6. ∆P'Q'R' is the image of ∆PQR after rotation about O at the given angle. Complete the table below: (a) (b) (c) X X X Y Y Y P Q P' R Q' R' (–90°) (+90°) 180° Q Q Q' Q' R' P' P' R' R R P O O O P O (– 90°)
GEOMETRY 305 (a) Pre-image Rotation Image (b) Pre-image Rotation Image P(1, 1) RO(–90°) P'( , ) P(1, 1) Ro (+90°) P'( , ) Q(4, 1) RO(–90°) Q'( , ) Q(2, 3) Ro (+90°) Q'( , ) R(2, –1) RO(–90°) R'( , ) R(3, –1) Ro (+90°) R'( , ) (c) Pre-image Rotation Image P(1, 3) RO(180°) P'( , ) Q(4, 2) RO(180°) Q'( , ) R(2, 1) RO(180°) R'( , ) 7. A(3, –3), B(5, 2) and C(4, 5) are vertices of ∆ABC. Find its image by stating coordinates and by graphing them after; (a) positive quarter turn about origin, (b) negative quarter turn about origin, (c) half turn about origin. 8. Repeat the question no. 5 for the triangle PQR with vertices P(5, 2), Q(3, –2), R(6, 4). 9. P(2, 1), Q(3, 3), R(5, 4) and S(6, 2) are vertices of a quadrilateral. Find the coordinates of the verticies of its image after; (a) positive quarter turn about origin, (b) negative quarter turn about origin, (c) half turn about origin. 10. P(1, 1), Q(4, –1), R(3, 1) and S(4, 3) are the vertices of a kite. Repeat the question no. the 7 for this.
306 The Leading Maths - 8 ANSWERS 6. (a) A(1, 1), B(4, 1), C(2, – 1); A'(1, – 1), B'(1, – 4), C'(– 1, – 2) (b) P(1, 1), Q(2, 3), R(3, – 1); P'(– 1, 1), Q'(– 3, 2), R'(1, 3) (c) K(1, 3), M(4, 2), N(2, 1); K'(– 1, – 3), M'(– 4, – 2), N'(– 2, – 1) 7. (a) A'(3, 3), B'(– 2, 5), C'(– 5, 4) (b) A'(– 3, – 3), B'(2, – 5), C'(5, – 4) (c) A'(– 3, 3), B'(– 5, – 2), C'(– 4, – 5) 8. (a) P'(– 2, 5), Q'(2, 3), R'(– 4, 6) (b) P'(2, – 5), Q'(– 2, – 3), R'(4, – 6) (c) P'(– 5, – 2), Q'(– 3, 2), R'(–6, – 4) 9. (a) P'(– 1, 2), Q'(– 3, 3), R'(– 4, 5), S'(– 2, 6) (b) P'(1, – 2), Q'(3, – 3), R'(4, – 5), S'(2, – 6) (c) P'(– 2, – 1), Q'(– 3, – 3), R'(– 5, – 4), S'(– 6, – 2) 10. (a) P'(– 1, 1), Q'(1, 4), R'(– 1, 3), S'(– 3, 4) (b) P'(1, – 1), Q'(– 1, – 4), R'(1, – 3), S'(3, – ) (c) P'(– 1, – 1), Q'(– 4, 1), R'(– 3, – 1), S'(– 4, – 3) Project Work 17.2 Draw a triangle on a graph. Rotate it through positive quarter turn about the origin and then its image angle is also rotated through 90° in anti-clockwise direction. Write the coordinates of the object triangle and its images in the table. What do you see on the coordinates of the object triangle and final image triangle? Write your conclusion. Prepare a report and present it in your classroom.
GEOMETRY 307 CHAPTER 18 BEARING AND SCALE DRAWING Lesson Topics Pages 18.1 Bearing 308 18.2 Scale drawing 312 What is direction compass? What is the full form of NEWS? What does SE represent? What is the meaning of N40°W? What does O80° represent? What is scale drawing ? Why is it needed? What is the meaning of 1:200 in scale drawing? What is the scale drawing of the object of the length 200 m if 2 cm = 15 m? What is the actual length of two places if its length in the map is 3 cm and scale drawing is 1:250000? S N W 40° E O N40°W WARM-UP
308 The Leading Maths - 8 At the end of this topic, the student will be able to: ¾ find the angle of bearing in the map. Learning Objectives 18.1 Bearing I. Introduction The figure alongside shows the eight directions in a compass card. From a north line to the North-East (NE) line; the angle between them is 45°. We write bearing in 3 digits. Hence the three figures bearing of NE measured from the north line is 045°. Now, write the bearing of the following compass directions in three figures. (a) East direction (E) → .... (b) South east direction (SE) → .... (c) South direction (S) → .... (d) South west direction (SW) → .... (e) West direction (W) → .... (f) North west direction (NW) → .... (g) North direction (N) → .... (h) North east direction (NE) → .... Now, we have the definition The bearing of a point A to a point B is the angle measured in clockwise direction from the north time. In the figure, the bearing A from B is 120°, while the bearing of B from A is the reflex angle 300°. The back bearing of B from A is worked out by adding 180° to the bearing of A from B. CLASSWORK EXAMPLES Example : 1 The bearing of a place Q from P is 045°. Find the bearing of P from Q. Solution: The bearing from P to Q is the angle NPQ = 045° given. Then the bearing of P from Q is the reflex angle N'QP. Here the reflex angle N'QP = 180° + 045° = 225°. S N W E NW NE SW SE N N' B A 120° N' N P 45° Q
GEOMETRY 309 Example : 2 What is the difference between the bearing of the given direction south and north east? Solution: Here Bearing of south = 180°, Bearing of NE = 045° ∴ Bearing between S and NE = 180° – 045° = 135° Example : 3 If an areoplance flies 20 km east then it changes the direction and flies 30 km south east. Also, it changes direction second time and flies 25 km south west. (a) Draw a rough sketch of the aeroplance course. (b) Find the bearing of the aeroplane from its final to starting position. Solution: (a) Drawing the rough sketches of the flying aeroplane accordingly the question. (b) When the aeroplane flies 20 km east from north, its bearing is 090°. When its direction changes to south east, its bearing is 135° and when its direction changes to south west its bearing is 225°, the bearing of SE from SW is 360° – 225° = 135° and the bearing of north from SW is 180° + 135° = 315°. EXERCISE 18.1 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) Define bearing? (b) What does the bearing 260° represent? (c) What is the meaning of 560° W? (d) If the bearing of P is 080° from P, then what is the bearing of Q from Q? NE W E S N 25 km 20 km SE E N N' N" N'" 30 km
310 The Leading Maths - 8 2. Draw a rough sketch to illustrate each of the following conditions. Mark the angle in your sketch. (a) From Kathmandu the bearing of Pokhara is 280°. (b) From Kathmandu the bearing of Dolakha is 055°. (c) From Rukum the bearing of Dolpa is 002°. (d) From Bajhang the bearing of Mugu is 100°. 3 Use the sketch to find the bearing of (a) C from A (b) C from D (c) A from C 4. What is the difference between the bearing of the given direction? (a) South and west (b) South and north east (c) North west and north east 5. What is the difference of the two angles between the given directions ? (a) North and east (b) North and south east (c) South west and north west 6. Use the sketch to answer the following questions. (a) What is the bearing B from C? (b) What is the angle between AC and due East? (c) What is the bearing for C from A? 7. If a bus drives 20 km east then it changes the direction and drives 20 km south east. Also, it changes the direction second time and drives 20 km south west. (a) Draw a rough sketch of the bus. (b) Find the bearing of the bus from its final to starting position. 8. A ship sails 6 km due north then changes its direction and sails 3 km due east and then changes its direction a second time and sails 3 km due south. (a) Draw a rough sketch of the ship's course. (b) Find the bearing of the ship from its starting to final position. N N' N'' C A D 45° 42° 40° N B C A N' N'' 35° 105° F D
GEOMETRY 311 9. The diagram alongside shows the course of an aeroplane immediately after leaving Tribhuvan International Airport. Find the bearing of the aeroplane in the position B from Airport T. 10. (a) The bearing of D from C is 212°, what is the bearing of C from D ? (b) If the bearing of H from G is 334°, what is the bearing of G from H ? (c) Ram and Sita are standing in a field. The bearing of Ram from Sita is 132°. What is the bearing of Sita from Ram ? ANSWERS 3. (a) 045° (b) 318° (c) 225° 4. (a) 090° (b) 135° (c) 090° 5. (a) 270° (b) 225° (c) 270° 6. (a) 245° (b) 125° (c) 325° 7. (a) Show to your teacher (b) 315° 8. (a) Show to your teacher (b) 135° 9. 279° 10. (a) 032° (b) 154° (c) 045° Project Work 18.1 Take a map of Nepal. Find the bearing of the capitals of provinces from Kathmandu by drawing direction lines. 61° 28° 127° T A B N N' N''
312 The Leading Maths - 8 At the end of this topic, the student will be able to: ¾ solve the problems related to scale drawing. Learning Objectives 18.2 Scale Drawing I. Introduction A scale drawing is an accurate picture of a real object. A scale factor is the ratio of one side of scale drawing to the corresponding side of the real object. The scale factor tells us how much larger or smaller the scale drawing is. Greater the scale factor larger the scale drawing and smaller the scale factor smaller the scale drawing. The corresponding angles of the real object and scale drawing remain the same. In the given figure, the real object and its scale drawing is drawn to a scale of 1:4. As the scale factor is fractional 1 4 , the scaled drawing is smaller than the real object. Measure the corresponding angles of the real object and the scaled drawing and check whether they are equal or not. Count the unit of area in both real objects and the scaled drawing and discover the relation between the area of the object figure and the scaled figure. Did you notice here, Area of the scaled figure Area of the object figure = 1 16 = 1 4 2 = Side of the scaled figure Side of the object figure 2 Now, we have the conclusion that, The scaled figure and object figure are similar. The areas of the object figure to the scaled figure are in the ratio of the squares of their corresponding sides. Scaled drawing Real Object
GEOMETRY 313 CLASSWORK EXAMPLES Example : 1 A scale drawing of a room is at the sale of 1:40. (a) If the room is 6 m long, what length does the scaled drawing represent this? (b) If the scaled length for the breadth of the room is 10 cm, find the actual breadth of the room. Solution: Let the figures represent roughly, 10 cm ? (y) ? (x) Scaled drawing 600 cm Actual drawing Window Door (a) As the scale factor is the ratio of scaled length to the object length, we have, x 600 = 1 40 or, x = 600 40 ∴ x = 15cm (b) 10 y = 1 40 or, y = 400 cm or, y = 4 m Note: The map scales are generally represented as 1:200000. This means 1 cm in the map represents 2,00,000 cm = 2 km in the real distance. Example : 2 An archived has made a scale drawing of the floor plan of a house using the scale 1:150 (in cm). Final the actual dimensions of (a) Lounge (b) Kitchen Directly you can work it out as: Length of the scale drawing = original length × scale ratio Directly you can work it out as: actual length = scale length × 1 scale ratio
314 The Leading Maths - 8 Lounge Hall Kitchen 2 cm 3.5 cm 5 cm 1.5 cm Solution: (i) The length of the lounge is (5 cm – 2 cm) = 3 cm × 150 cm 1 cm = 450 cm = 4.5 m [ 1 m = 1 100 cm] Breadth of the lounge = 3.5 cm × 150 cm 1 cm = 525 cm = 5.2 m \ The dimension of the lounge is 4.5 m by 5.25 m. (ii) Length of the kitchen is 2 cm × 1.5 m 1 cm = 3.0 m [ 150 cm = 1.5 m] Breath of the kitchen is 1.5 cm × 1.5 m 1 cm = 2.25 m ∴ The dimension of the kitchen is 3 m by 2.25 m. Example : 3 The diagram is a scale diagram of a room. The scale used is 1:200. Measure the length and breadth of the given diagram. What is the size of the real room? Solution: We measure the dimensions of the room, Length = 4 cm ∴ The actual measurement of length of the room is 4 cm × 200 1 = 800 cm = 8 m
GEOMETRY 315 Breadth = 3 cm ∴ The actual measurement of breath is 3 cm × 200 1 = 600 cm = 6 m ∴ The dimension of the room is 8 m by 6 m. Example : 4 ABCD is a rectangular field in which, AD = 32 m and DC = 49 m. ABCD is drawn by using a scale of 1 cm to 5 m. i.e. in the ratio of 1:500 (i) How long should AD and DC be as the scale drawing ? (ii) On the drawing, AB is 12 cm long. How long is the side AB of the real field ? Solution: Here, in the rectangular field ABCD, AD = 32 m and DC = 49 m scale ratio = 1:500 Now, ∴ The measurement of AD in drawing is 32 m × 1 500 = 3200 cm 500 = 6.4 cm and the measurement of DC in drawing is 49 m × 1 500 = 4900 cm 500 = 9.8 cm ii) Since measurement of AB is 12 cm. ∴ Field measurement of AB is 12 cm × 5 m 1 cm = 60 m [ 500 cm = 5 m] ∴ The side AB of the real field is 60 m. EXERCISE 18.2 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is scale drawing? Write its one use? (b) What does the ratio of scale drawing mean? (c) A scale drawing of a room is at a scale of 1:30. If the room is 600 cm long, what is its length on the scale diagram?
316 The Leading Maths - 8 (d) A scale drawing of a house is at a scale of 1:20. The real height of the house is 10 m. What is its height of the scale drawing? (e) A model of truck is at a scale of 1:200. If the model is 4 cm long, what is the length of the real truck? (f) A toy elephant is at a scale of 1:90. If the toy is 3 cm high, what is the height of the real elephant? 2. A playing field WXYZ is to be drawn to scale. (a) Find what the drawn lengths of WX and XY should be if the scale is 1 cm to 8 m. (b) If the measure of XZ is 10 cm, what is the real length of XZ ? 3. By using a scale of 1:5 and centimeter squared paper, make a scale drawing of each shape. (a) 15 cm 10 cm (b) 35 cm 20 cm (c) 25 cm 25 cm 4. By using a scale of 1:500 and centimeter squared paper, make an accurate scale drawing of each field. (a) (b) 5. Use a scale of 4: 1 and centimeter squared paper, make a scale drawing of each shape. (a) (b) (c) Z W X Y 72 m 96 m 50 m Rectangle 30 m 70 m 75° 105° 60 m 40 m 60 m
GEOMETRY 317 6. A map showing two towns A and B which are 6 km apart has a scale of 1:20000 (a) How many centimeters are there in 6 km ? (b) What length on the map represents the distance between the two towns B and C 4.5 km apart? 7. The map below has a scale of 1:40000 (in cm). City Hall Market Bus Station College Hospital School Find the actual distance in between, (a) the hospital and city hall. (b) the school and bus station. (c) the buy station and college 8. An aeroplane flies 120 km on a bearing of 130°. If then it changes its course and flies 220 km on a bearing of 035°. (a) Draw a scale diagram by using the scale 1 cm to 13 cm and protractor. (b) How far is the aeroplane due east of its stay point? (c) How far is the aeroplane due north of its stay point? 9. Use the map of the scale 1:200000 to work out, (a) distance between Taplejung to Kathmandu. (b) distance between Pokhara to Kanchanpur. 10. This map is drawn to a scale of 1 cm to 20 kilometers. State the distance, the nearest kilometers from Kathmandu to: (a) Muglin (b) Pokhara (c) Trishuli 220 km 120 km 35° N NN 130° Trisuli Kathmandu Pokhara Muglin
318 The Leading Maths - 8 11. The drawing below is the plan of Hari’s house, using a scale of 1:200. Dining Room Bath Room Lounge Kitchen Shower Sidi Sidi 7 cm 3 cm (a) By using scale or ruler what is the size on this plane of: i. the dinning room. ii. the kitchen? (b) What is the actual dimension of: i. the dinning room. ii. the kitchen ? ANSWERS 1. (a) 20 cm (b) 50 cm (c) 8 m (d) 2.7 m (e) 12 cm, 9 cm (f) 80 m 6. (a) 30 cm (b) 22.5 cm 7. (a) 2.8 km (b) 2 km (c) 2.912 km Project Work 18.2 Measure the length and breadth of your bedroom. Find the length and breadth of the room using appropriate scale drawing and draw the map. Prepare a report and present it in your classroom.
GEOMETRY 319 1. A transversal cuts two lines as shown in the figure (a) Find the value of a and b. (b) Find the value of 2c. (c) For what values of c, the two lines will be parallel? Justify with reason. 2. A triangle ABC is shown in the figure with AN // BC. (a) Write the relation of co-interior angle while a pair of parallel lines are intersected by a transversal. (b) Find the value of x and y. (c) Verify experimentally that the relation of interior angles of a triangle by making two different size of triangles. 3. ABCD is a parallelogram with diagonal AC. A line BC produced to E and ∠DCE = 60°. (a) Write the relation of ∠ADC and ∠DCE. (b) Find the value of ∠BAD. (c) Prove that: ΔABC ≅ ΔADC. (d) Construct a parallelogram ABCD, in which AB = 4 cm, BC = 3.2 cm and ∠ABC = 60o by using a compass. 4. A shape of square is given alongside. (a) Write the formula for finding the interior angle of a regular polygon. (b) Find the interior angle of regular quadrilateral. (c) What should be the exterior angle of regular pentagon? (d) Construct a square having side 4 cm. 5. In the figure, AB//CD and AC = CE. (a) Find the interior angle of an equilateral triangle. (b) From which axioms, ΔABC is congruent to ΔDEC? (c) Prove that BC = CD. (d) Are all the congruent triangles similar? Justify. 3b 2c 5b 55° a B x y C A N 40° 72° C E 60° B A D B E D A C MIXED PRACTICE–V
320 The Leading Maths - 8 6. Four equilateral triangles and one square having sides equal in length are given below. (a) Which solid object can be made from the given four triangles. (b) Which solid object can be made from the given four triangles and a square? Draw that solid. (c) Draw the geometric shape of cylinder. 7. In the XY plane, a line AB is drawn. (a) Write the co-ordinate of A and B. (b) Draw a horizontal line from A on right and a vertical line from B on down, name C for the intersecting point. (c) Find the distance of AC and BC. (d) Find the distance of AB by using Pythagoras formula. 8. A triangle with coordinate is shown in the co-ordinate plane. (a) Find the distance between (2, 3) and (– 3, – 1). (b) Prove that the given triangle of the given graph is a right angled triangle. (c) Naming A for (2, 3), B for (2, –1) and C for (– 3, – 1) then find the coordinates of the image of ∆ABC under reflection in y-axis. (d) Are the object and its image after reflection congruent? Justify. 9. Plot the co-ordinate of A, B and C of the ΔABC in a graph, where, A = (3, –3), B = (5, 2) and C = (4, 5). (a) Find the coordinates of the vertices of its image after positive quarter turn about the origin. y B 5 -5 5 -5 4 -4 4 -4 3 -3 3 -3 2 -2 2 -2 1 -1 1 -1 A x y 4 -4 4 -4 3 -3 3 -3 2 -2 2 -2 1 -1 1 -1 (2, –1) (2, 3) (–3, –1) x
GEOMETRY 321 (b) Find the coordinates of the vertices of its image after translation by T 3 4 . (c) Find the coordinates of the vertices of its image after reflection in x- axis and draw its image on the same graph. (d) Are object and its image after rotated + 90o about the origin congruent? Justify. 10. A bearing model is given alongside. (a) Write a bearing of B from A. (b) Find the bearing of A from B. (c) Find the length of AC. 11. An equilateral triangle is given alongside. (a) Define tessellation with example. (b) Draw a tessellation from an equilateral triangle. (c) How many symmetric lines can be drawn on the given triangle? (d) How many symmetric lines can be drawn on the regular polygon? A B C N M 150° 150° 60° 12 km 5 km ANSWERS 1. (a) 125°, 25° (b) 75° (c) 27.5° 2. (a) 68°, 40° 3. (b) 120° 4. (b) 90° (c) 72° 5. (a) 60° 7. (c) 9 units, 4 units (d) 9.85 units 8. (a) 6.4 units (c) A'(– 2, 3), B'(–2, – 1), C'(3, – 1) 9. (a) A'(3, 3), B'(– 2, 5), C'(– 5, 4) (b) A'(6, 1), B'(8, 6), C'(7, 9) (c) A'(3, 3), B'(5, – 2), C'(4, – 5) 10. (a) 60° (b) 240° (c) 13 km
322 The Leading Maths - 8 1. In the given figure, EF is the transversal of the pair of lines AB and CD. (a) Find the value of x. [1] (b) Find the value of y. [1] (c) Write the relation between the lines AB and CD. [1] 2. (a) Draw two triangles ABC in different shape and size. [1] (b) Measure each angle of both triangles and show the results in the table. [2] (c) Write the conclusion about the sum of angles of ∆ABC. [1] (d) If ∠A = 30° and ∠C = 54° in ∆ABC, find the measure of ∠B. [1] 3. ABCD is a rhombus with diagonals AC and BD. (a) Write the relation between AC and BD. [1] (b) If AC = 16 cm and BD = 12 cm, find the length of AD. [2] (c) Construct a rectangle PQRS, in which PQ = 5.3 cm and QR = 4.2 cm by using a compass. [2] 4. (a) In the given figure, ∆MNL ≅ ∆RST. Fins the value of x. [2] N L M R T S (2x + 30)° 55° 65° (b) Find the measure of an interior angle of regular heptagon. [2] A B C D y x 130° 50° D A B C FM : 23 Time : 40 Min. CONFIDENCE LEVEL TEST V GEOMETRY Attempt all the questions.
GEOMETRY 323 (c) Which solid is formed by the given net. [1] 5. (a) Write the co-ordinates of A and B. [1] Y Y' X' X A B (b) Find the length of AB of the graph. [2] (c) Find the bearing from B to A in the given figure. [1] (d) Write the type of tessellation of the given figure. [1] A 50° N N' B
324 The Leading Maths - 8 Work Periods - 10 (Th. + Pr.) COMPETENCY classify, present and describe the data . CHAPTERS 19. Statistics LEARNING OUTCOMES take and give from pie chart and construct pie chart from the given data, find the mean, median and mode of the individual data. STATISTICS UNIT VI 17.03% 20.98% 20.84% 17.55% 5.8% 8.49% 9.28% Koshi Madesh Bagmati Gandaki Lumbini Karnali Sudur Pashchim Nepal Population by Province
STATISTICS 325 CHAPTER 19 STATISTICS Lesson Topics Pages 19.1 Pie Chart 326 19.2 Arithmetic Mean 330 19.3 Median and Mode 333 What is circle ? What is the central point of the circle called? What is the name of OA in the given circle? Tell the relation between OA, OB and OC. What is the name of ∠AOB? How many degrees are in the circle around its centre O? How many types of representation of data are there? Which is greater or smaller number between 25 and 37? What is the ascending or descending order of the numbers 3, 7, 4, 9, 8, 10, 15, 12? Which is the fifth number in the data 5, 7, 9, 12, 15, 18, 20, 24? How many numbers of item are in the data 3, 5, 8, 10, 15, 20, 38, 47, 45? Which is the middle number in the data 8, 12, 13, 17, 19, 25, 28? What are row data and individual data? Can you arrange the given data in ascending order? Arrange 7, 8, 10, 10, 8, 6, 7, 8, 20, 15, 8, 9 in ascending order. Which item is the most repeated in the above data? What is called most the repeated item in the above data? B A C O WARM-UP
326 The Leading Maths - 8 19.1 Pie-Chart At the end of this topic, the student will be able to: ¾ draw the pie-chart and line graph from the given data. Learning Objectives I. Introduction We use pie chart to display information that can be counted or grouped. It is useful to compare frequency of different things with each other. However they do not tell the actual frequencies of the categories. This pie chart shows how Rajiv spends his 24 hours period in each day. The area of each sector of the pie chart depends on the size of the category of stand for. The given pie-chart shows the information of time table of Rajiv. How to convert each sector into time ? Discuss. How much time did he spend for doing homework ? How much time did he spend for watching TV ? 60° 90° 120° 60° 30° Watch T.V Do homework Sports and News Paper Sleep Study at School Index II. Draw of a pie chart (a) Use the frequency table from your original raw data. (b) Add all the frequencies. (c) Calculate the size of the angle for the slices of pie chart. For this you can use the formula. Angle at the centre of the slice = Frequency of one category Total frequency × 360° You must be sure that the sum of the angles made by the slices at the centre must be up to 360°. Remark: If the given data is given in the percentage form, we calculate the angle at the centre each slice = Percentage of each category 100% × 360°
STATISTICS 327 CLASSWORK EXAMPLES Example : 1 Table below shows the quantities of favourite of fruits among students in a school. Apple 25 Bananas 30 Orange 35 Strawberries 40 Mangoes 50 Total 180 Draw a pie chart to represent the above information. Solution : Here, Calculation of the angle at the centre of the slice, we have Apples : 25 180 × 360° = 50° Bananas : 30 180 × 360° = 60° Oranges : 35 180 × 360° = 70° Strawberries : 40 180 × 360° = 80° Mangoes : 50 180 × 360° = 100° Total = 360° Now, we have drawn the pie chart below, Apples Bananas Oranges Strawberries Mangoes Index
328 The Leading Maths - 8 EXERCISE 19.1 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is pie chart? (b) Write the formula to find the measure of the central angles. (c) What is the measure of central angle covered 120° units chart of 720 units? 2. Table below gives the votes obtained by different parties in an election in a village. Parties CPNUML NC CPN Maoists CPN United Sadbhavana No. of voters 4500 4000 1500 2500 500 Draw a pie chart to represent this information. 3. Petrol supplied from different petrol pumps during Terai bandha in a week is given below. Petrol pumps Pump A Pump B Pump C Pump D Pump E Quantity of petrol 30000 l 45000 l 40000 l 50000 l 20000 l Draw a pie chart to represent this information. 4. Number of tourists that visited Nepal in last 5 years is given below. Year 2018 2019 2020 2021 2022 No. of tourists 40000 55000 60000 30000 20000 Draw a pie chart to represent this information. 5. A dice is thrown 36 times. The results are shown in the table below. Digit on Dice 1 2 3 4 5 6 Frequency of getting digit 4 8 5 6 7 6 Draw a pie chart to represent the above data. 6. The population of Nepal is are shown in the table below. Province Koshi Madhesh Bagmati Gandaki Lumbini Karnali S. Pashchim Population (%) 17% 21% 21% 8% 18% 6% 9% Present the above information in a pie chart.
STATISTICS 329 7. Read the pie chart given below and answer the questions that follow. Now answer the following questions (a) Which is the most popular subject? (b) Which is the least popular subject? (c) If a total of 720° students were interviewed, how many liked: i. Maths ii. Science iii. English iv. Social Studies v. Population (d) What percentage of the total favour for i. Maths ii. Social Studies iii. Science 8. A Departmental store surveyed among 200 customers to ask what was their favourite topping to go with chips. The results are presented below in a pie chart. Answer the following questions. (a) Complete the table below. Topping Don't like chips Salt and vinegar Ketchup Mayonnaise Curry sauce Gravy Cheese Percentage 5% No. of customers 10 (b) How many customers chose either cheese or ketchup ? (c) How many more customers chose curry sauce than salt and vinegar ? (d) Do more customers refer curry sauce and gravy, or salt and vinegar and ketchup? ANSWERS Consult with your teacher. Project Work 19.1 Record your expenses on a week. Present it in a pie chart. Prepare a report and present it in your classroom. Social Studies 85° Population 60° English 45° Science 70° Maths 100° Popularity of school subjects 5% 20% 16% 23% 11% 21% 4% Do not like chips Salt and vinegar Ketchup Mayonnaise Curry sauce Gravy Cheese
330 The Leading Maths - 8 19.2 Arithmetic Mean At the end of this topic, the student will be able to: ¾ calculate the arithmetic mean of the ungrouped data. Learning Objectives I. Introduction ACTIVITY - 1 What is the sum of the amounts 23, 24, 25, 26, 27 that five boys have? If they distribute it equally among them, how much does each boy get? How many students are there? Definition The arithmetic mean of the given observations or data is defined as : Mean = Sum of the given observation or data No. of observation. II. Calculation of Arithmetic Mean for Ungrouped Data Arithmetic Mean for Individual Data If an individual data has N observations like as x1 , x2 , x3 ........ xn , then its arithmetic mean is given by, Arithmetic mean (x) = Sum of N observation Total No. of observation = x1 + x2 + x3 + ........ +xn N = ∑x N CLASSWORK EXAMPLES Example : 1 Find the arithmetic mean of the marks obtained by 10 students. 15, 18, 22, 25, 30, 32, 40, 48, 60, 70 Solution : Here; Sum of marks (ΣX) = 15 + 18 + 22 + 25 + 30 + 32 + 40 + 48 + 60 + 70 = 360 No. of Students (N) = 10 Now, Mean ( X ) = ΣX N = 360 10 = 36
STATISTICS 331 EXERCISE 19.2 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is average? (b) Define arithmetic mean? Write its formula. (c) Which formula is used to calculate the mean of discrete data? (d) What is the mean of the data 1, 2, 3, 4, 5, 6, 7, 8, 9, 10? (e) What is the mean of the data 10, 20, 30, 40, 50, 60, 70? 2. Find the arithmetic mean for the set of data given below. (a) Marks obtained by 10 students: 65, 69, 72, 77, 81, 83, 85, 90, 92, 96 (b) Runs made by a cricket team in 10 innings : 8, 12, 10, 7, 12, 18, 24, 20, 22, 30 (c) Life of electricity bulbs in 1000 hours : 3.1, 1.2, 2.4, 3.5, 4.8, 7.0, 6.3, 7.2, 4.5 (d) Height of the students in cm 121, 120, 119, 122, 125, 130, 128, 127, 140, 135, 129, 137, 148, 124, 132, 140 (e) The price of mountain bikes in (Rs.100) 200, 180, 220, 190, 250, 225, 395, 199, 290, 251, 250, 260, 270, 350, 255, 360, 240, 355 3. A book seller sold the books for ten days for the following amount in Rs. 10250 12521 35252 42837 12542 62510 52152 25327 28352 37284 (a) Find the average sale (b) Estimate the sale for a month.
332 The Leading Maths - 8 4. The daily wage of the workers in a factory for 30 works is given below. 120 140 135 120 125 115 130 145 150 160 125 132 140 160 140 145 135 150 128 130 135 160 140 150 128 122 135 165 120 145 (a) Find the average daily wage of a worker. (b) Estimate the total expense on wage of the workers for 3 months if the total number of workers employed in the factory is 200. 5. The daily earning of a family for 10 days is given below. Rs. 150 Rs. 225 Rs. 528 Rs. 400 Rs. 250 Rs. 175 Rs. 240 Rs. 180 Rs. 222 Rs. 440 (a) Find the average income in a day. (b) Estimate the monthly earning. (c) Work out the saving of the family expense in a month when 75% of the earning is expended. 6. (a) If ∑x = 420, N = 17 + P and mean (x) = 20, find the value of P. (b) In a data, of mean (x) = 3, ∑fx = 150 + 2a and N = 15 + a, find the value of 'a'. ANSWERS 2. (a) 8.1 (b) 16.3 (c) 0.04 (d) 129.81 (e) 26.333 3. (a) Rs. 31902.70 (b) Rs. 957081 4. (a) Rs. 137.5 (b) Rs. 2475000 5. (a) Rs. 281 (b) Rs. 8430 (c) Rs. 2107.50 6. (a) 4 (b) 5 Project Work 19.2 List out the marks obtained in the unit test of your all subjects. Calculate the arithmetic mean of your marks obtained. Also, find the percentage of your marks obtained of all subjects. Compare the arithmetic mean and percentage of your marks obtained. Write your conclusion about it. Prepare a report and present it in your classroom.
STATISTICS 333 At the end of this topic, the student will be able to: ¾ find the median of the given ungrouped data. ¾ find the mode of given ungrouped data. Learning Objectives 19.3 Median and Mode I. Introduction When data are arranged in order of size in ascending or in descending order, one value lies in the middle if the data are odd in number. For example, in the collection 3,4, 5, the number 4 lies in the middle. Hence 4 is the value of middle term. If the data are even in number, there are two values in the middle. For example, in the collection 1, 2, 4, 6 the numbers 2 and 4 lie in the middle. In such case, the middle term is the average of 2 and 4. Hence, the value of the middle term = 2 + 4 2 = 3. Hence, we have the definition. The median of the statistical data is the value or term that lies exactly in the middle of the distribution. If the data are odd in number, there is a single median. If the data are in even number, the single median is worked out by taking average of the two middle values or terms. II. Estimation of Median in Individual Data The median for the individual data may be estimated as follows; CLASSWORK EXAMPLES Example : 1 Consider two data; (a) 2, 5, 8, 7, 6, 3, 4 (b) 10, 12, 9, 8, 16, 17, 18, 20 Solution : (a) Arranging the data in ascending order we get 2, 3, 4, 5, 6, 7, 8 The number 5 lies in the middle because about and below 5 there lie equal number of data Hence, median = 5
334 The Leading Maths - 8 (b) Arranging the data in ascending order we get 8, 9, 10, 12, 16, 17, 18, 20 Here, there is not a single value at the middle. There are 12 and 16 as the middle values. Hence, median = 12 + 16 2 = 14 We can use a formula for working out the median. (a) When data are odd in number then Median = Value of N + 1 2 th item. N + 1 2 th item = 7 + 1 2 th item = 4th item Counting either from top or from bottom, the 4th item is 5. (b) When data are in even number, then there are 2 middle values. Hence data Median is the average of N 2 th item and N + 2 2 th items. In example (b), N = 8 Therefore, N 2 th item = 8 2 th = 4th item = 12 N + 2 2 th item = 8 + 2 2 th = 5th item = 16 And average of N 2 th and N + 2 2 th item = 12 + 16 2 = 14. III. Introduction to mode The mode is the most common (or the data point that appears most often) in a set of data. It can be found by putting the data into an ordered list and seeing which data point occurs most often. How to find the mode? Step (1): Put the data into an ordered list. Step (2): Check that you have got the same number of data points.
STATISTICS 335 Step (3): The mode is the data point which is the most common. For example; Find the mode of 3, 6, 4, 3, 2, 4, 7, 8, 6, 3, 9 Step (1): Put the data into an ordered list. This gives us: 2, 3, 3, 3, 4, 4, 6, 6, 7, 8, 9 Step (2): Check the number of data points in both lists is the same. Both lists have 11 data points. Step (3): The mode is the number which occurs most often. Answer: The mode is 3. CLASSWORK EXAMPLES Example : 2 Find the mode of 0.6, 0.3, 0.4, 0.2, 0.4, 0.7, 0.6, 0.1, 0.4, 0.9 Solution: Here, arranging the given data in ascending order, we get 0.1, 0.2, 0.3, 0.4, 0.4, 0.4, 0.6, 0.6, 0.7, 0.9 Since the mode is the number which occurs most often so, the mode is 0.4. EXERCISE 19.3 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is median? Define it. (b) Which formula is used to calculate the median of ungrouped data? (c) What is the median of 3, 7, 8, 9, 12, 20? (d) What is the median of 20, 30, 40, 50? (e) What is mode? (f) What is the mode of 23, 24, 23, 22, 25, 23? (g) What is the mode of 11, 12, 13, 15, 12, 14, 13?
336 The Leading Maths - 8 2. Circle ( ) the correct answers. (a) Median is the term of the given data. i. average ii. middle iii. first iv. last (b) Which formula is used to calculate the median of the data? i. N 2 ii. ∑X N iii. W –1 2 iv. W + 1 2 (c) What is the median of the data 10, 12, 15, 18, 17, 20, 25? i. 17 ii. 15 iii. 18 iv. 25 (d) The mode of the given individual data is ................... i. middle item ii. average item iii. first item iv. most repeated item (e) Which is the middle value in the data 2, 3, 5, 7, 3, 7, 4, 3, 1? i. 1 ii. 2 iii. 3 iv. 4 (f) What is the mode of the data 13, 14, 16, 14, 17, 18, 16? i. 13 ii. 14 iii. 16 iv. both (i) and (ii) 3. Find the median for the data given below. (a) Shoe size : 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7 (b) Height (cm) : 130, 120, 122, 128, 132, 140, 128, 125, 131, 134 (c) Weight (kg) : 52, 48, 42, 40, 41, 51, 50, 56, 46, 58, 57, 47 (d) Length (cm) : 15.2, 12.8, 10.5, 11.2, 17.6, 15.8, 12.3, 13.5, 14.2, 17.5, 18.1 (e) Marks : 20, 25, 15, 18, 22, 16, 19, 14, 23, 24, 27, 26, 15 (f) Wage (Rs.) : 140, 110, 120, 150, 160, 135, 125, 130, 140, 145, 155, 135.
STATISTICS 337 4. Find the mode of the following data: (a) 3, 2, 4, 5, 6, 3, 5, 8, 4, 2, 3, 4, 3 (b) 24, 34, 43, 25, 34, 56, 24, 43, 24, 35, 24, 34 (c) 24, 15, 18, 20, 18, 22, 24, 26, 18, 26, 24 (d) 8, 11, 9, 14, 9, 15, 18, 6, 9, 10 (e) 123, 145, 132, 132, 123, 145, 123, 156, 123, 134, 123 5. Find the mode from the data given below. (a) 5, 13, 13, 22, 6, 18, 6, 23, 10, 8 (b) 40, 35, 22, 18, 18, 40, 36, 45, 18 (c) 5.8, 6.2, 9.5, 5.8, 6.2 7.9, 5.8, 12.6 (d) 9.2, 9.9, 9.2, 9.2, 9.8, 9.5, 9.9, 9.2 (e) 1.25, 1.23, 1.23, 1.24, 1.45, 1.23, 1.56, 1.23 ANSWERS 3. (a) 5 (b) 129 (c) 49 (d) 47.5 (e) 20 (f) 137.5 4. (a) 3 (b) 24 (c) 24 (d) 9 (e) 123 5. (a) 18 (b) 27 (c) 37 (d) 0.7 (e) 0.33 Project Work 19.3 Measure the height of 10 friends with you. Arrange them in ascending order. Whose height is exactly middle? Find it. Prepare a report and present it in your classroom. Weight 15 students of your class and arrange in ascending order. Find which student has modal weight. Prepare a report and present it in your classroom.
338 The Leading Maths - 8 1. Tourists that visited Nepal in last 5 years are as given below. Years 2003 2004 2005 2006 2007 Number of tourists 40000 50000 55000 60000 30000 (a) Draw a pie chart to represent this information. (b) Find the average number of tourists during 5 years. 2. Marks obtained in Mathematics by 10 students in a final exam of grade 8 are as follows: 36, 39, 42, 47, 41, 42, 45, 40, 42, 46 (a) Write the model marks of the students. (b) Is the average mark greater than 40? Justify with calculation. (c) Are the mean and median equal? 3. Ram's spend time during 24 hours is given below in pie chart. 60° 90° 120° 60° 30° Watch TV Do homework Sports and News Paper Sleep Study at School Index (a) How much times does Ram spend for doing homework? (b) Is the average time for spending on five activities 4 hours in a day? Justify. 4. Total weights (kg) of 10 students studied at grade 8 are as follows: 50, 48, 42, 40, 41, 51, 50, 56, 42, 50 (a) Write the model weights of the students. (b) Prove that the average weight of 10 students is 47 kg. (c) If 56 is removed from the above data, what will be the average weights of students? MIXED PRACTICE–VI
STATISTICS 339 5. A dice is thrown 30 times and the result is tabulated as follows. Digits on Dice 1 2 3 4 5 6 Frequency of getting each number 7 2 6 3 4 8 (a) Draw a pie chart to represent this information. (b) Which digit on a dice repeated more times? Write it. 6. Look at the pie chart showing the number of books in the library of different subjects. Answer the following equations. (a) How many books are there in the library of the English ? (b) How many more books are there of English than of Science ? (c) Which subject books are maximum in the library ? (d) Which subject books are minimum in the library ? (e) How many total books are there in the library ? ANSWERS 1. (b) 47000 2. (a) 42 (b) 42 (c) Yes 3. (a) 4 hrs (b) No 4. (a) 50 (c) 41.4 kg 5. (b) 6 200% 300% 400% 100% French Science English Maths
340 The Leading Maths - 8 FM : 15 Time : 40 Min. CONFIDENCE LEVEL TEST IV Statistics 1. The number of students in the classes from 6 to 10 in the table below. Class 6 7 8 9 10 Number of students 135 165 150 150 120 (a) Draw a pie chart to represent the above information. [2] (b) What is the mode the above table. [1] 2. The marks obtained by six students of the roll number from 1 to 6 of the class 8. 75, 90, 60, 45, 30, 60 (a) Draw a pie chart to represent the above data. [2] (b) What is the average makes of these six students. [1] 3. Observe the given figure. (a) What represents the given figure. [1] (b) If there are 41 pears in the shop, how many total fruits are there? Find it. [1] (c) How many bananas are there in the shop? Find it. [1] 4. Observe the given pie chart of the favourable subjects of 180 students. (a) Which is the most favourable subject ? How many students liked it? [1] (b) Which is the least favourable subject ? How many students liked it? Find it. [1] (c) How many more students liked more favourable subject than least favourable subject? Find it. [1] 5. The height of 9 students studied at the grade 8 are as follows. 145 cm, 155 cm, 150 cm, 142 cm, 160 cm, 140 cm, 142 cm, 155 cm, 150 cm (a) Which formula is used to find the median of an individual data ? [1] (b) What is the median height among the students? Find it. [1] (c) If a student of the height 145 cm is removed from the above data, what will be the average height of the students? Find it. [1] 20.5% 25.5% 19.5% 10.5% 11.5% 12.5% Apples Kiwi Banana Mango Pears Orange Mathematics 108° 45° 54° 72° 81° Social Science English Nepali Attempt all the questions.
STATISTICS 341 1. The sets A and B are defined as, A = {Prime numbers less than 10} and B = {Even numbers from 1 to 10} (a) Are the sets A and B are overlapping or disjoint sets? [1] (b) Make any six subsets from set A. [2] 2 A girl deposits Rs. 25000 for 3 years at the rate of interest is 10%. (a) What is the formula to find the interest? [1] (b) Calculate the amount with interest. [2] (c) Convert 25000000 in scientific notation. [2] 3. An article of the price Rs. 40000 after allowing 10% discount and leving 13% VAT was sold. (a) Write the formula to find the discount percentage. [1] (b) Find the discount amount. [1] i. Rs. 40 ii. Rs. 400 iii. Rs. 4000 iv. Rs. 4500 (c) What is the VAT amount? [2] (d) Find the ratio between discount amount and VAT amount. [2] 4. A student bought 20 note copies for Rs. 2000. (a) Find the CP of one note copy. [1] (b) Convert the CP of 20 note copies in quinary numbers system. [1] (c) How much selling price should be taken to get Rs. 25 profit per copy? [1] 5. In the given figure, the base (AD) = 12 cm and height (AB) = 8 cm. (a) Find the area of ∆ABC and ABCD. [2] (b) Find the ratio of the area of ∆ABC and ABCD. [1] 12 cm 8 cm A B C D PRACTICE QUESTION SET Class – 8 Subject: Mathematic F.M. : 50
342 The Leading Maths - 8 6. The radius of a circle is r cm. (a) Write the area of the circle in term of r. [1] (b) Find the area of the circle if r = 14 cm. p = N 2 [1] 7. (a) Write the value of (a + b)0 . [1] (b) Factorize : x2 + x [1] (c) If x + y = 4 and xy = 3, then find the value of x2 + y2 . [2] 8. The area of rectangular field is (x2 + 12x + 32) m2 and breadth (x + 4) m. (a) Find the length of the field. [2] (b) If x = 4 m, then find the actual area of the field. [2] 9. The sum of two numbers 32. If one of them is more than another number by 4. Find these numbers. [2] 10. In the given figure AB||CD. (a) Which is the corresponding angle of ∠QSB. [1] i. ∠ ASR ii. ∠SRD iii. ∠BSR iv. ∠QSB (b) What is the sum of ∠ASP and ∠CRQ ? [1] (c) Construct a regular pentagon having side 6 cm. [3] (d) Write the formula to find the measure of an exterior angle of a regular polygon. [1] 11. In the given ∆ABC, the side BC is produced up to D where AC = BC. (a) Find the values of x and y. [2] (b) Write two distinct properties of square and rectangle. [2] 12. (a) P(4, 3), Q(7, 3) and R(4, 3) are vertices of ∆PQR. Reflect its vertices in y-axis and plot the coordinates of the vertices of ∆PQR and its image on the same graph. [3] (b) Define the bearing and sketch the bearing of 065°. [2] S A R C P Q D B A 110° y x B C D
STATISTICS 343 SPECIFICATION GRID SN Area Total working hours Knowledge Understanding Application Higher ability Total number of items Total number of question Total Marks Number of items Marks Number of items Marks Number of items Marks Number of items Marks 1 Sets 10 1 1 1 1 2 3 1 1 5 2 3 2 Statistics 10 3 3. Arithmetic 45 2 2 3 4 3 5 2 3 10 3 14 4. Mensuration 15 1 1 1 1 1 2 1 1 4 1 5 5. Algebra 30 2 2 1 2 2 4 1 2 6 3 10 6. Geometry 50 2 2 2 4 2 6 2 3 8 3 15 Total 160 8 8 8 12 10 20 7 10 33 12 50 Class 6 - 8 (2078) Full marks : 50 Time : 2 hours MATHEMATICS While constructing the question paper, the weightage of the specifications for knowledge, understanding, application and high efficiency in each area should be met. But in the numerical scale, it can decrease up to 2 points. Questions should be constructed by giving references. Each question can contain more than one compound level sub-question. While creating questions of application and higher ability level, there may be questions related to the subject matter of other areas besides the correlation area. The questions should be moderated so that the contents of all the sub-areas within each area are proportionately included. Note :
344 The Leading Maths - 8 Sample for Project Work - 1.2 Objectives: To represent the set in set notation. To identify the types of sets. Procedure: 1. The subject teacher may ask the students to perform the task a small group of five students. 2. The students will collect the materials from the geometric box. 3. The students will represent them in set notation in various methods. 4. The student will mention them in types of sets. Methods A) Collecting and Listing Materials from Geometric Box: Different types of materials are collected from the geometric box and listed them by using the first or last letter of the name of materials. Let the set be denoted by any capital letter and represent in various methods. B) Comparison of Materials: The name of the collection materials are compared between friends. Identify who has more, who has less, who has equal number of materials and who has the same materials. C) Construction of Subsets: Construct the subsets by taking at most 5 elements of the set. If we take more than 5 elements, it will be difficult to construct the subsets. Discuss on the proper and improper subsets from the whole subset. Presentation: 1. Introduce me or every members of a group. 2. Present the whole work step by step and turn by turn. Conclusion: 1. From the project work, I/we conclude that I/we learned the set can be represented in various methods. 2. From the project work, I/we conclude that I/we learned the sets can be categorized into different types that are easily identified. 3. Thank you for listening to me/us.