Geometry GEOMETRY 301 19.2 Reflection At the end of this topic, the students will be able to: ¾ reflect the given figures in the coordinate plane. Learning Objectives Introduction A motion or change or transformation may or may not take place according to some rule or rules. In mathematics, the word Transformation means any motion or shift or change that occurs according to some given rule or rules. A transformation may be geometric or algebraic. Reflection Rotation Translation Enlargement A transformation can be described in terms of coordinates. We shall consider only two cases: reflection and rotation. In these two cases also, we shall take up very simple cases for coordinate treatment. Reflection A light ray falling on a plane mirror is reflected back. A vertical ray falling on a horizontal mirror is reflected vertically upwards and along the same line. Suppose light rays coming from a point source at a certain distance strike a plane mirror normally. To a person standing in front of the mirror, it appears to come from a point behind the mirror. The later point is observed to be at the same distance from the plane as the original point or source from which the light ray came. Reflection thus associates a point with another point determined by perpendicular incidence on a plane surface like a mirror. Of the two points, the fist one is called the object point and the second one as the image point relative to the plane or mirror. Such a plane is usually represented by a straight line. The line or the plane is called the axis of reflection. Reflection is thus a correspondence or matching of a point with a point on the other side of a line or plane such that
302 The Leading Mathematics - 7 i. both of them are at the same distance from the line or plane and ii. both of them lie on the same line drawn perpendicular or normal to the given line or plane from the original point. It preserves angles, shape and size of the geometric figure or object. However, the reversal of sides can be observed: Left becomes right and right becomes left if the mirror is vertical; but top becomes bottom and bottom becomes top if the mirror is horizontal) i. An instructive experimental demonstration of reflection can be performed in the following way: Draw a triangle ABC in a plane sheet of paper. Draw a line XY not intersecting the triangle. Fold the paper along the line. Prick through the vertices of the triangle. Then, there will be corresponding marks or holes (A', B' and C', say) on the other fold beneath it. Open fully the paper sheet. Join the corresponding pairs of points (or holes). We can then see that (a) the pairs of points A and A′, B and B′, C and C′ are on opposite sides of the line XY (the axis of reflection), (b) each pair of points lie on a line perpendicular to the line XY, (c) the points A and A′, B and B′, C and C′ are at equal distances from the line XY, and (d) line segments AA′, BB′ and CC′ are parallel, The line reflection just described gives rise to a mirror image. It is not difficult to prove the two triangles are congruent. So, their shape and size are preserved. ii. Let's now see what happens when a second line reflection takes place on a line parallel to the first axis. Reflection in a plane is quite simple and easily understandable. In order to obtain the reflected image of any plane figure whatever, we (a) choose a line as the axis of reflection, (b) drop a perpendicular to the axis of reflection and then find an associated point corresponding to every point of the figure, B B′ A A′ X Y C C′
Geometry GEOMETRY 303 (c) produce the perpendicular to a point whose distance from the axis is equal to the length of the perpendicular in (ii). Now, we discuss reflection in some standard lines. (a) Reflection in the x-axis or in the line y = 0 Suppose P is any point not on the x-axis. To locate its image, drop PM perpendicular on the x-axis and produce it to P′(x', y') making MP = MP′. The four possible cases are shown below (x, y > 0): P′(x, – y) P(x, y) M X Y O P′(– x, – y) P(-x, y) M X Y O Obviously, the coordinates of the image or reflected points are given by Fig. (a) x′ = x and y′ = –(y) = –y, i.e., (x, y) → (x, –y) Fig. (b) x′ = (–x) and y′ = –(y) = –y i.e., (–x, y) → (–x, –y) From the above observations, we conclude that when reflection in the x-axis takes place, the sign of the y-coordinate changes and nothing else. Thus, any point P(x, y) is transformed into the point P′(x, –y). In symbols, Reflection in x-axis (x, y) → (x, –y) (b) Reflection in the y-axis or in the line x = 0 Suppose P is any point not on the y-axis. To locate its image, drop PN perpendicular on the y-axis and produce it to P' making NP = NP’. We may proceed as in the case of reflection in the x-axis. But, it is not necessary. We may simply consider the case in which P lies in the first quadrant. Suppose P(x, y) is any point in the first quadrant and P'(x', y') is its image on the y-axis. The point P' lies in the second quadrant and its coordinates will be (–x, y), that is there will be change in the sign of the x-coordinate. In symbols, Reflection in y-axis (x, y) → (–x, y)
304 The Leading Mathematics - 7 CLASSWORK EXAMPLES Example 1 Reflect the points A(1, 3), B(−4, −3), C(−7, 8) and D(8, −9) on (a) x-axis (b) y-axis Solution: Here, the given points are A(1, 3), B(–4, –3), C(–7, 8) and D(8, –9). Now, (a) Reflection of the x-axis: Equations of reflection on the x-axis are x′ = x and y′ = − y. Any point (x, y) changes to (x, − y). ∴ A (1, 3)→A′ (1, −3), B (− 4, − 3) → B′ (− 4, 3) C (− 7, 8) → C′ (− 7, − 8) D (9, −8) → D′ (9, 8) (b) Reflection on y-axis: Equations of reflection on the y-axis are x′ = − x and y′ = y. Any point (x, y) changes to (−x, y). A (1, 3)→A′ (−1, 3) B (−4, −3) → B′ (4, −3) C (−7, 8) →C′ (7, 8) D (9, −8) → D′ (−9, 8) Example 2 Observe the vertices A(−2, 2), B(2, 2) and C(2, 6). (a) Plot these vertices on the graph paper. (b) Reflect the ∆ABC on x-axis. (c) Show the image on a graph paper. Solution: (a) Plotting the vertices A(–2, 2), B(2, 2) and C(2, 6) on the graph. (b) Now, reflection on x-axis. Any point (x, y)→(x, −y) ∴ A (−2, 2) →A′ (−2, −2) B (2, 2) →B′ (2, −2) C (2, 6) →C′ (2, −6) (c) Representing the image on graph. Y' O Y X' X A'(–2, –2) C'(2, –6) C'(2, 6) B'(2, –2) A(–2, 2) B(2, 2)
Geometry GEOMETRY 305 Example 3 The end-points of the line segment OA are O(0, 0) and A(3, 4). (a) Reflect OA on the x-axis. (b) Reflect its image OA’ again on the y-axis. (c) Show the two images on the same graph paper. Solution: Here, the end-points of an arrow OA are O(0, 0) and A(3, 4). Now, (a) Reflection of OA on the x-axis Any point (x, y)→(x, −y) ∴ O (0, 0) →O (0, 0) A (3, 4) →A' (3, −4) (b) Reflection of OA' on the line y-axis. Any point (x, y) →(– x, y) O (0, 0) →O′ (0, 0) A′ (3, −4) →A′' ( −3, −4) (c) Showing the above reflections on the same graph. Y' Y X' X O A(3, 4) A''(–3, – 4) A'(3, – 4) EXERCISE 19.2 Your mastery depends on practice. Practice like you play. 1. Reflect the given figure on the given line : (a) A (b) B C X' X Y Y' x-axis
306 The Leading Mathematics - 7 2. Reflect the following points on the x-axis and write the coordinates of their images: (a) (1, 2) (b) (–2, 3) (c) (– 4, – 3) (d) (a, b) 3. What are the coordinates of the image of the following points under reflection in the line x = 0? (a) (2, – 4) (b) (–1, –3) (c) (0, 4) (d) (– p, q) 4. Observe the shape in the given graph. (a) Write the image of a point P(x, y) after reflection on the line x = 0. (b) Write the coordinates of the given shape of the graph. (c) Reflect the given shape in the x-axis on the same graph. (d) Write the coordinates of the vertices of the image shape. 5. (a) The points A(4, 5), B(−2, 3), C(6, −7) and D(−8, −9) are reflected in x-axis. Write the coordinates of image of these points. Also, represent the above relation in graph paper. (b) The vertices of ∆ABC are A(−1, 2), B(4, 1) and C(1, 5) respectively. Write the coordinates of vertices of image of ∆ABC obtained on reflecting it in x-axis. Also, draw the image on graph paper. 6. (a) (7, 3), (3, 0), (0, −4) and (4, −1) are the vertices of a rhombus PQRS. Reflect it on y-axis and write the coordinates of the images. Also, draw the image on graph paper. (b) The end-points of an arrow OA are O(0, 0) and A(3, 3). Reflect OA on the x-axis. Reflect its image O'A' again on the y-axis. Show the two images on the same graph paper. ANSWERS 2. (a) (1, – 2) (b) (– 2, – 3) (c) (– 4, 3) (d) (a, – b) 3. (a) (– 2, – 4) (b) (1, – 3) (c) (0, 4) (d) (p, q) 5. (a) A'(4, – 5), B'(– 2, – 3), C'(6, 7), D'(– 8, 9) (b) A'(– 1, – 2), B'(4, – 1), C'(1, – 5) 6. (a) P'(– 7, 3), Q'(– 3, 0), R'(0, – 4), S'(– 4, – 1) (b) O'(0, 0), A'(3, – 3); O''(0, 0), A''(– 3, – 3) Y' Y X' X O P Q R
Geometry GEOMETRY 307 Geometry 19.3 Translation At the end of this topic, the students will be able to: ¾ translate the given object in vertical and horizontal lines. Learning Objectives A translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a oneto-one correspondence between two sets of points or a mapping from one plane to another.) In a translation transformation, all the points in the object are moved in a straight line in the same direction. The size, the shape and the orientation of the image are the same as that of the original object. Same orientation means that the object and image are facing the same direction. For example, We describe a translation in terms of the number of units moved to the right or left and the number of units moved up or down. Move the object 2 units to the right and 4 units up. Translation in horizontal and vertical lines In the adjoining graph, when the line segment AB moves 3 units on right side, what will be changed in its images? What are the coordinates of A and B of AB ? After moving it 3 units right, the coordinates of A' and B' of the image A'B' are shown in the table below. Coordinates of Preimage Movement of Point on right Coordinates of Images A(......, .......) 5 A' (......, .......) = A' (......, .......) B(......, .......) 5 B' (......, .......) = B' (......, .......) In general, if we translate a point P(x, y) in the direction of ‘a’ units on horizontally right or left, the image P(x', y') will be P'(x + a, y), a ∈ R. i.e., P(x, y) P'(x + a, y) A A' A'' C C' C'' B B'' B' A B A' B' X Y' O Y X'
308 The Leading Mathematics - 7 In the adjoining graph, when the line segment AB moves 3 units on up side, what will be changed in its images? What are the coordinates of A and B of AB ? After moving it 3 units up, the coordinates of A' and B' of the image A'B' are shown in the table below. Coordinates of Preimage Movement of Point on upside Coordinates of Images A(......, .......) 5 A' (......, .......) = A' (......, .......) B(......, .......) 5 B' (......, .......) = B' (......, .......) In general, if we translate a point P(x, y) in the direction of ‘b’ units on horizontally up or down, the image P'(x', y') will be P'(x + a, y), a ∈ R. i.e., P(x, y) P'(x, y + b) Translation on Horizontally and Vertically The translation can be represented by a column vector as a b . The top number represents the right and left movement. A positive number means moving to the right and a negative number means moving to the left. The bottom number represents up and down movement. A positive number means moving up and a negative number means moving down. In the figure alongside, triangle TRI with vertices T(2, -1), R(4, 3) and I(-3, -2) is being translated to triangle T′R′I′ by the vector –5 2 . Then the coordinates of the vertices of the image ∆T′R′I′ are T′(-3, 1), R′(-1, 5) and I′(-8, 0). T(2, – 1) → T′(– 3, 1) = T′(2 + (– 5), – 1 + 2) R(4, 3) → R′(– 1, 5) = R′(4 + (– 5), 3 + 2) I(– 3, – 2) → I′(– 8, 0) = R′(– 3 + (– 5), – 2 + 2) A B A' B' X Y' O Y X' R′(-1,5) I′(-8,0) T(2,-1) R(4,3) T′(-3,1)
Geometry GEOMETRY 309 Let P(x, y) be a point, then the coordinates of its image under the translation through the vector a b will be P′(x + a, y + b). i.e., P(x, y) P′(x + a, y + b) a b EXERCISE 19.3 Your mastery depends on practice. Practice like you play. 1. Draw the image of the following figures under the given instructions: B C C (a) 4 units right. Q R P (b) 3 units down. K N L M (c) 2 units left and 3 units down 2. Write down the coordinates of the vertices of the figures in the following graphs and copy them in the graph paper. Translate it by the given vector. Also, find the coordinates of the vertices of their images: (a) X Y O I L J K (b) X Y O B C D A (c) X Y O S P Q R Vector – 4 0 Vector 0 3 Vector 4 – 4
310 The Leading Mathematics - 7 3. Translate the following points by the given translation vector: (a) A(4, – 2) by 3 1 (b) B(4, 3) by –2 0 (c) C(– 2, 3) by 4 –1 4. (a) Plot the vertices P(3, – 2), Q(2, 0) and R(– 1, 1) of ∆PQR on the graph and find the coordinates of the vertices of its respective image under the translation vector 3 –2 . Also, draw the given geometric shape and its image on the same graph. (b) Plot the vertices A(– 3, 1), B(– 2, – 3), C(3, – 4) and D(1, 3) of the quadrilateral ABCD on the graph and find the coordinates of the vertices of its respective image under the translation vector –2 3 . Also, draw the given geometric shape and its image on the same graph. (c) Translate ∆ABC with vertices A(2, 5), B(2, 0) and C(– 2, 3) by the translation vector –1 3 . Write down the coordinates of the vertices of the image of ∆ABC and represent the above translation in the graph. 5. Observe the shape in the given graph. (a) Write the image of a point P(x, y) after translation by the vector a b . (b) Write the coordinates of the given shape of the graph. (c) Translate the given shape 2 units down on the same graph. (d) Write the coordinates of the vertices of the image shape. ANSWERS 3. (a) (7, – 1) (b) (2, 3) (c) (2, 2) 4. (a) P'(6, – 4), Q'(5, – 2), R'(2, – 2) (b) A'(– 5, 4), B'(– 4, 0), C'(1, – 1), D'(– 1, 6) (c) A'(1, 8), B'(1, 3), C'(– 3, 6) Y' Y X' X O P Q R
Geometry GEOMETRY 311 CHAPTER 20 Bearing and Scale Drawing Lesson Topics Pages 20.1 Bearing 312 20.2 Scale Drawing 316 What is direction of compass? What is the full form of NEWS? What does SE represent? What is the meaning of N40°W? What does 080° 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? WARM-UP 3.32 cm 6.81 cm 6.81 cm 4.76 cm
312 The Leading Mathematics - 7 20.1 Bearing At the end of this topic, the students will be able to: ¾ write and draw the compass bearing of the point or place. Learning Objectives Compass and Its Points A compass is used to find a direction or bearing. The four main directions of a compass are known as cardinal points. They are north (N), east (E), south (S) and west (W). Sometimes, the half-cardinal points of north-east (NE), north-west (NW), south-east (SE) and south-west (SW) are shown on the compass. A directional compass as shown below. It is used to find a direction or bearing . The four main directions of a compass are known as cardinal points. They are north (N), east (E), south (S) and west (W). Sometimes, the half-cardinal points of north-east (NE), north-west (NW), south-east (SE) and south-west (SW) are shown on the compass. The above compass shows degree measurements from 0° to 360° in 10° intervals with: north representing 0° or 360° east representing 90° south representing 180° west representing 270° When using a directional compass, hold the compass so that the point marked north points directly away from you. Note that the magnetic needle always points to the north. The 16-point wind compass is used to determine the names of the compass directions. The cardinal directions are north (N), east (E), south (S), west (W), at 90° angles on the compass rose. The ordinal (or intercardinal) directions are northeast (NE),southeast(SE),southwest (SW) and northwest(NW), formed by bisecting the angle of the cardinal winds. The name is merely a combination of the cardinals it bisects. S N E WNW ENE ESE SSW WSW SSE NE SE SW NW NNW NNE W NE SE SE NW S N W E S N E WNW ENE ESE SSW WSW SSE NE SE SW NW NNW NNE W NE SE SE NW S N W E
Geometry GEOMETRY 313 The eight principal winds (or main winds) are the cardinals and ordinals considered together, that is N, NE, E, SE, S, SW, W, NW. Each principal wind is 45° from its neighbour. The principal winds form the basic eight-wind compass rose. The eight half-winds are the points obtained by bisecting the angles between the principal winds. The half-winds are north-northeast (NNE), east-northeast (ENE), east-southeast (ESE), south-southeast (SSE), south-southwest (SSW), westsouthwest (WSW), west-northwest (WNW) and north-northwest (NNW). Notice that the name is constructed simply by combining the names of the principal winds to either side, with the cardinal wind coming first, the ordinal wind second. The eight principal winds and the eight half-winds together yield a 16-wind compass rose, with each compass point at a 22 1 2 ° angle from the next. Bearing Bearing is used to give direction in aviation. It is defined as a positive angle from 0o to 360o measured clockwise with respect to the north. 34° A bearing of 34° North 133° A bearing of 133° North 300° A bearing of 300° North A bearing is an angle, measured clockwise from the north direction. Below, the bearing of B from A is 025 degrees (note 3 figures are always given). The bearing of A from B is 205 degrees. A B 25° 205° N N'
314 The Leading Mathematics - 7 EXERCISE 20.1 Your mastery depends on practice. Practice like you play. 1. Write the directions in the compass below and write down their bearings. 2. Find the compass bearing of the following directions: (a) NE (b) SW (c) ES (d) WN (e) NW (f) NEN (g) ENE (h) SEN (i) NSN (j) SWN 3. Find the compass bearing of the given points in the following diagram by using a protractor. (a) (b) (c) (d) (e) (f) N P N A N Q N D N R N C
Geometry GEOMETRY 315 4. Draw the diagrams using a protractor to show the following compass bearing: (a) N45o E (b) N70o E (c) E30o N (d) S60o W 5. Write the values in 3-figure bearing of the following compass bearings: (a) S30o W (b) E70o S (c) N54o E (d) W60o N 6. Draw the diagrams using a protractor to show the following 3-digit bearing: (a) 035o (b) 125o (c) 240o (d) 315o 7. Study the given map of Nepal and write the answers of the following questions: Ilam Nepalganj Lumbini Rara Lake Pokhara Kathmandu Sagarmatha Kanchanjanga (a) Consider Kathmandu as the centre, write the bearing of the following places by using a protractor. i. Rara Lake ii. Lumbini iii. Ilam iv. Sagarmatha (b) Write the bearing of the following places considering Sagarmatha as the centre by using a protractor. i. Kathmandu ii Nepalganj iii. Pokhara iv. Lumbini ANSWERS 2. (a) 045° (b) 225° (c) 135° (d) 315° (e) 315° (f) 022 1 2 ° (g) 067 1 2 ° (h) 112 1 2 ° (i) 000° (j) 247 1 2 ° 5. (a) 210° (b) 160° (c) 054° (d) 330°
316 The Leading Mathematics - 7 20.2 Scale Drawing Suppose the length and breadth of the bedroom are 8 m and 6 m respectively. Similarly, suppose the length of the mold is 5 microns (µ). 1 mm = 1000 µ Can we draw the actual length and breadth of the bedroom and the actual length of the mold in the notebook? We cannot draw the actual size of large figures or objects or very tiny figures or objects in the paper. To remove these problems, we can draw the models of big or small figures or objects by reducing or enlarging their actual size by using suitable scale. 4 cm 3 cm 8 m 6 m 0.005 mm 3 cm Bedroom Mold 248 Allied Mathematics-7 5.8 (B) Scale Drawing Suppose the length and breadth of the classroom are 8 m and 6 m respectively. Similarly, suppose the length of the mold is 5 microns (µ). 1 mm = 1000 µ Can we draw the actual length and breadth of the classsroom and the actual length of the mold in the notebook? We cannot draw the actual size of large figures or objects or very tiny figures or objects in the paper. To remove these problems, we can draw the models of big or small figures or objects by reducing or enlarging their actual size by using suitable scale. 4 cm 3 cm 8 m 6 m 0.005 mm 3 cm Classroom Mold Hence, the process of reduction or enlargement of the actual figure or objects on the paper is called scale drawing. In the above drawing, Length of figure of classroom = 4 cm Breadth of figure of classroom = 3 cm Length of actual classroom = 8 m Breadth of actual classroom = 6 m What is the ratio of the length and breadth of the figure of classroom and actual classroom? The ratio of lengths = 4 cm 8 m = 4 cm 8 × 100 cm = 1 200 = 1:200 The ratio of breadths = 3 cm 6 m = 3 cm 6 × 100 cm = 1 2000 = 1:200 Hence, the scale drawing is 1:200. This means the length or breadth of the classroom is 200 times reducing in the figure on the paper i.e. 1 unit in the figure is equal to 200 units in the actual object. This constant ratio of the scale drawing is called the scale factor. Class room Mold on orange 248 Allied Mathematics-7 5.8 (B) Scale Drawing Suppose the length and breadth of the classroom are 8 m and 6 m respectively. Similarly, suppose the length of the mold is 5 microns (µ). 1 mm = 1000 µ Can we draw the actual length and breadth of the classsroom and the actual length of the mold in the notebook? We cannot draw the actual size of large figures or objects or very tiny figures or objects in the paper. To remove these problems, we can draw the models of big or small figures or objects by reducing or enlarging their actual size by using suitable scale. 4 cm 3 cm 8 m 6 m 0.005 mm 3 cm Classroom Mold Hence, the process of reduction or enlargement of the actual figure or objects on the paper is called scale drawing. In the above drawing, Length of figure of classroom = 4 cm Breadth of figure of classroom = 3 cm Length of actual classroom = 8 m Breadth of actual classroom = 6 m What is the ratio of the length and breadth of the figure of classroom and actual classroom? The ratio of lengths = 4 cm 8 m = 4 cm 8 × 100 cm = 1 200 = 1:200 The ratio of breadths = 3 cm 6 m = 3 cm 6 × 100 cm = 1 2000 = 1:200 Hence, the scale drawing is 1:200. This means the length or breadth of the classroom is 200 times reducing in the figure on the paper i.e. 1 unit in the figure is equal to 200 units in the actual object. This constant ratio of the scale drawing is called the scale factor. Class room Mold on orange Hence, the process of reduction or enlargement of the actual figure or objects on the paper is called scale drawing. In the above drawing, Length of figure of bedroom = 4 cm Breadth of figure of bedroom = 3 cm Length of actual bedroom = 8 m Breadth of actual bedroom = 6 m What is the ratio of the length and breadth of the figure of bedroom and actual classroom? The ratio of lengths = 4 cm 8 m = 4 cm 8 × 100 cm = 1 200 = 1:200 The ratio of breadths = 3 cm 6 m = 3 cm 6 × 100 cm = 1 2000 = 1:200 Hence, the scale drawing is 1:200. This means the length or breadth of the bedroom is 200 times reducing in the figure on the paper i.e. 1 unit in the figure is equal to 200 units in the actual object. This constant ratio of the scale drawing is called the scale factor. What is the scale factor for the figures of mold? Bedroom Mold on orange
Geometry GEOMETRY 317 CLASSWORK EXAMPLES Example 1 The given figure represents the shape of rectangular garden. If the scale factor of the figure and the garden is 1:5000, find the actual length and breadth of the garden. Solution: Given, Length of the rectangular figure = 3 cm Breadth of the rectangular figure = 2.5 cm Scale factor = 1:5000 ∴ Actual length of the rectangular garden = 3 cm × 5000 = 15000 cm = 150 m ∴ Actual breadth of the rectangular garden = 2.5 cm × 5000 = 12500 cm = 125 m Example 2 In the adjoining map, K, P, N, B and E represent the different places. Find the actual distance between (i) K and N (ii) P and B (iii) N and E by using scale factor 1 cm : 25 km. Solution: Here, the scale factor = 1 cm : 25 km (i) The distance between K and N in the map = 2.1 cm ∴ The actual distance = 2.1 × 25 km = 52.5 km (ii) The distance between P and B in the map = 3.2 cm ∴ The actual distance = 3.2 × 25 km = 80 km (iii) The distance between N and E in the map = 3.5 cm ∴ The actual distance = 3.5 × 25 km = 87.5 km EXERCISE 20.2 Your mastery depends on practice. Practice like you play. 1. (a) The length and breadth of a rectangular room are 10 m and 8 m. Draw the figure of the room in your notebook by taking the scale of 1 cm : 5 m. (b) A rectangular garden has length 250 m and breadth 200 m. Draw the rectangular figure in your copy by taking the scale factor 1:10000 (in cm). 3 cm 2.5 cm K P N B E
318 The Leading Mathematics - 7 (c) The horizontal distance between Kathmandu and Rara lake is 275 km. By taking a scale of 1 cm : 25000 m, draw the line in your copy. 2. (a) If the distance between Kathmandu and Ilam in the map is 8 cm. What is the actual arial distance between them (scale factor 1:120000). (b) The length of Great Asian Highway is 4.1 cm in a map. Find the actual distance of the highway by taking scale of 1 : 1000000000. 3. Find the actual dimensions of the following objects by taking the scale factor 1:520. (a) (b) (c) (d) 4. Study the given map of Nepal and find the distance between the following places: (Scale 1:5500000) Pokhara Hetauda Janakpur Biratnagar Bhaluwang Surkhet Godawari ANSWERS 1. (a) 2 cm by 1.6 cm (b) 2.5 cm by 2 cm (c) 11 cm 2. (a) 960 km (b) 141000 km 3. (a) 10.40 m by 7.80 m (b) 20.80 m by 15.60 m (c) 10.40 m (d) 21.84 m 2 cm 1.5 cm Room 4 cm Garden 3 cm Tower 4.2 cm 2 cm Tree (a) Kathmandu and Surkhet (b) Hetauda and Pokhara (c) Bhaluwang and Biratnagar
Geometry GEOMETRY 319 Read, Understand, Think and Do 1. Two lines AB and CD are intersected at point O. (a) Measure the angles ∠AOC, ∠BOC, ∠AOD and ∠BOD by using a protractor and fill the result as shown below in the table. Angles ∠AOC ∠BOC ∠AOD ∠BOD Degree (b) Are the angles ∠AOD and ∠BOC or ∠AOC and ∠BOD equal? Write and conclude the result. (c) What is the sum of ∠AOC and ∠BOC or ∠AOD and ∠BOD? Write and conclude the result. (d) Two angles ∠AOC and ∠BOC are either adjacent angles or supplementary angles? Justify with reason. 2. Draw a line AB with length 5 cm as shown in the figure. (a) Construct the angles of 75o from A. (b) Draw the angle of 105o from B. (c) Give the name of C in intersected point. Is the sum of three interior angles 180o ? Calculate it. 3. In the given figure, ∠ XOY = x and ∠ WOZ = y are given. (a) Are x and y equal? Give reason. (b) Measure the angles ∠XOY, ∠XOW, ∠WOZ and ∠YOZ by using a protractor and fill the result by making a table. (c) Is the sum of all the angles 360°? Justify with reason. A B Z W y O x Y X MIXED PRACTICE–V O C A B D
320 The Leading Mathematics - 7 4. A triangle ABC having size AB = 4.5 cm, ∠A = 60° and B = 30° is given alongside. (a) Look at the figure and construct it in your copy by using a compass and ruler. (b) Can we write AB2 = AC2 + BC2 ? Write with reason. 5. A ladder rests against a wall of height 8 ft. If the foot of the ladder is 6 ft away from the foot of the wall. (a) Write the formula for Pythagorean relation. (b) Find the length of the ladder. (c) If the height of the wall is 6 ft, what type of triangle would be made? 6. A right-angled triangle PQR is presented in the coordinate plane. (a) Write the coordinate of point P and R. (b) Reflect the line PR in x- axis. (c) Find the distance of PQ and QR. (d) Find the length of PR by using Pythagoras theorem. 7. In the co-ordinate plane. XX' is horizontal line and YY' is vertical line with center O. (a) In which quadrant points do A, C and D lie? (b) Write the co-ordinates of C and D. (c) Are the length of AD and BC equal? (d) Reflect AD and BC in the y-axis. (e) Translate B in 2 units right and 2 units down. 4.5 cm 60° C A B 30° O X Y Q R P O A B C D X' X Y Y' 8 ft 6 ft
Geometry GEOMETRY 321 8. A bearing model is given alongside. (a) Write a bearing of B from A. (b) Find the bearing of A from B. 9. An equilateral triangle is given alongside. (a) Define tessellation with example. (b) Draw a tessellation from an equilateral triangle. (c) How many line symmetry can be drawn from the given triangle? (d) Are all equilateral triangles congruence in each other? Justify. 10. A solid figure of tetrahedron is given alongside. (a) Draw the skeleton model of tetrahedron. (b) How many vertices, faces and edges of tetrahedron are them. (c) Is the Euler's formula satisfied? Justify it. A B C M 150° 150° 60° N'' N' N 12 km 8 km Tetrahedron ANSWERS 4. (b) Yes 5. (b) 10 ft. (c) Isosceles right triangle 6. (a) (3, 6), (7, 3) (b) (3, – 6), (7, – 3) (c) 3 units, 4 units (d) 5 units 7. (b) (– 1, – 1), (2, – 2) (c) 4 units, 4 units 8. (a) 060° (b) 240° 9. (c) 3 (d) No 10. (b) 4, 4, 6
322 The Leading Mathematics - 7 1. Study the given information and write the answers of the following questions. (a) Draw a line segment of the length 5.4 cm and name it. [1] (b) Construct the angles 90° and 60° on its end points by using a compass. [2] (c) Are the arms of the arms that make the above mentioned angles ? Name the intersecting point. [1] (d) What type of triangle do you find ? Name it. [1] 2. Observe the given figure and answer the following questions. (a) Write any one property of a parallelogram. [1] (b) Find the value of 'x'. [2] (c) Find the length of the diagonal AC. [1] (d) In which condition does it become a rhombus. [1] 3. (a) In which condition do two circles congruent ? [1] (b) Draw the skeleton model of a cuboid. [2] (c) Justify the Euler's formula in a square pyramid. [2] 4. (a) Plot the points P(0, 0), Q(3, 0) and R(0, 3) on the graph paper. Write the name of shape when these points are joined. [2] (b) Which types of symmetry are there in the given figure ? Write the number of symmetry lines on it. [1] (c) Find the coordinates of the image of a point (3, 4) reflecting on the x-axis. [1] (d) Find the compass bearing of the direction WN. [1] M L K N O x + 10 5x + 2 Attempt all the questions. FM : 20 Time : 40 Min. CONFIDENCE LEVEL TEST V GEOMETRY
Geometry STATISTICS 323 COMPETENCY Collection and presentation of data. CHAPTERS 21. Presentation of Data LEARNING OUTCOMES take and give the information from line graph. construct the line graph from the given data. draw multiple bar graph from the data. STATISTICS UNIT VI
324 The Leading Mathematics - 7 CHAPTER 21 Presentation of Data Lesson Topics Pages 21.1 Line Graph 325 21.2 Multiple Bar Graph 333 Can you read the scales of the graph ? What are positive and negative parts in the graph? What does the x-axis represent in the simple bar graph? What does the y-axis represent in the simple bar graph? What do the pillars represent in the simple bar graph? Can you plot the coordinates on the graph? What types of figure do you get when two points are joined? WARM-UP
Geometry STATISTICS 325 21.1 Line Graph At the end of this topic, you will be able to: ¾ draw the line graph from the given data. Learning Objectives Review on Data The word Statistics ordinarily means information expressed in terms of numbers. Such numerical information are known as data. Data are collected and compiled. They may be discrete like individuals or continuous like time. Age of a person, his height, his weight or the like can be specified numerically or as data. But, we cannot do so with his religion, his nationality or the like. Data may be collected from the original place of inquiry or sources. Such data are primary and others are secondary. Data thus obtained are usually crude, ungrouped and unsystematic. They become more usable when grouped or classified. Grouped or classified data look more comfortable when presented in the form of tables, graphs, charts, and diagrams. Now, let's see how this can be done: (a) Data Collection i. Primary Data The marks of 20 students in Statistics are 35, 59, 65, 43, 78, 59, 37, 78, 71, 75, 35, 78, 65, 78, 71 43, 59, 78, 35, 78. This is also called raw data. ii. Secondary Data The marks of the first ten students in Statistics from the school record are 78, 78, 78, 78, 78, 78,75, 71, 71, 65, 65, 59, 59, 59, 43, 43, 37, 35, 35, 35. We use a letter such as x to denote any one of such data or information. (b) Tabulation Representation We can put the data in the form of a simple table. In such tables, repeated data may be grouped together. In other words, frequency of a particular item or value can be written down. Such a table is known as frequency table or frequency distribution table. The following is one such table:
326 The Leading Mathematics - 7 Frequency Distribution of Marks of 20 Students Marks of Students (x) Tally Marks Number of Students (Frequency f ) Cumulative Frequency (cf ) 35 37 43 59 65 71 75 78 /// / // /// // // / //// / 3 1 2 3 2 2 1 6 3 3 + 1 = 4 4 + 2 = 6 6 + 3 = 9 9 + 2 = 11 11 + 2 = 13 13 + 1 = 14 14 + 6 = 20 Total 20 This table has individual value for frequency. Therefore, it is called frequency table of ungrouped data. It is also called discrete data. Introduction to Line Graph Data presented in the form of tables are easy to handle and easy to understand. But, a pictorial representation is much more appealing and eye-catching. Graphs, diagrams and charts are good visual aids even to the layman. In this section we shall briefly discuss them. Common information such as variations of temperatures during a day, rise and fall of prices of goods, changes in the population size, marks scored by the first five students in a test, etc. can be represented quite nicely by means of lines: straight, curved or pieces of line-segments joined end to end. Such a pictorial representation of data is called a line graph. A line graph is a plane figure. Ordinarily, two lines at right angles to each other are drawn on a plane sheet of paper. Points are marked off on each line at equal intervals. The numbers or items on which another items depend are represented by points on the horizontal line, and the other numbers or items by the points on vertical line. The meeting point of the horizontal and vertical lines through these points represents the corresponding pair of information (data or items). If we join every consecutive pair of such points by a line-segment, we get the complete picture (or line-graph) of the whole data. Some-times, a smooth line (or a line
Geometry STATISTICS 327 without any sharp corner) also serves the purpose. Now, we give some examples of line graphs. The population of Kirtipur (every ten years) is given in the following table. Population of Kirtipur Year Population 1950 50,000 1960 1,00,000 1970 2,00,000 1980 2,50,000 1990 3,00,000 2000 3,50,000 The line graph shows the population of Kirtipur from 1950 to 2000. 400,000 350,000 300,000 250,000 200,000 150,000 100,000 50,000 0 Population 1950 1960 1970 1980 1990 2000 Years In the graph the horizontal (axis show) line in years and the vertical areas shows the population. What is the population of 1960 ? Can you estimate the population of 1965 ? What is it ? It is about 1,50,000. We use line graphs to see the changing trend on certain periods of line. The line between the points shows the trend.
328 The Leading Mathematics - 7 Making Line Graphs Before drawing a graph, we must make a few decisions. 1. What title will you use for his graph ? 2. What scale will you use ? 3. Will you round the remembers ? If so, to what place ? 4. How will you label each a rust? CLASSWORK EXAMPLES Example 1 Draw a line graph showing the marks obtained in English by the first five students in a class. Student First Second Third Fourth Fifth Marks (E) 30 65 50 40 55 Solution: Drawing a line graph showing the marks obtained in English by the first five students in a class as given below; 0 10 20 30 40 50 60 70 1st 2nd Students Line Graph Marks 3rd 4th 5th Example 2 Draw a line graph showing the following monthly wages and the number of workers as follows: Monthly wages (in Rs.) 3250 3500 3750 4000 4250 Total Number of workers 25 30 50 40 35 180 Solution: Drawing a line graph showing the given monthly wages and the number of workers as follows:
Geometry STATISTICS 329 0 10 20 30 40 50 60 70 3250 3500 Wages in Rupees Line Graph No. of Workers 3750 4000 4250 Example 3 Draw the line-graphs showing the marks obtained by the first five students in English and Mathematics. Student First Second Third Fourth Fifth Marks (E) 30 65 50 40 55 Marks (M) 50 30 40 50 65 Solution: Line-graphs showing the marks obtained by the first five students in English and Mathematics. Here the two lines are drawn in the same diagram. In such a case the trend of difference between them can be easily seen. 0 10 20 30 40 50 60 70 1st 2nd Students English Mathematics Special line Graph Marks 3rd 4th 5th EXERCISE 21.1 Your mastery depends on practice. Practice like you play. 1. (a) In which axes should the marks obtained and the number of students be shown to draw a line graph of a discrete data? (b) In which axes should variables and frequencies be shown to draw a line graph of a discrete data?
330 The Leading Mathematics - 7 (c) The given line graph shows the number of teens in a small town who have cell phones based on the age groups. Which age group of teens has more? (d) In which day did the temperature show minimum in the city in the given line graph ? 2. A wildlife biological study on counting the number of hippopotamus and recorded the information on a line graph given below and answer the following questions based on the graph: (a) In 2010, what was the number of hippopotamus? (b) How many hippos were in 2007? (c) In which year did the number reach 30? (d) Which year recorded the minimum number of hippopotamus? (e) How many more hippopotamus were in the year 2008 than in 2006? 3. George works as a salesman in an authorized car showroom. He records the number of cars sold in five days (Monday to Friday) in a line graph. Study the graph and answer the questions. (a) On which day was the maximum number of cars sold? (b) How many cars were sold on Wednesday? (c) Which day had the minimum sales of cars? (d) How many more cars were sold on Tuesday than on Monday? (e) How many total cars sold on these five days? 229 Q. No. 1(c) Q. No. 1(d) Small town Teens With Cells Phones Age in years No. of Teens 0 200 400 600 800 13 14 15 16 17 18 19 0 20 40 60 80 1 2 3 4 5 6 Day Temperatures in New York City Degrees in Fahrenheit 273 341 430 590 530 642 43 53 50 57 59 67 0 4 8 12 16 20 24 28 32 36 2005 2006 2007 2008 2009 2010 Hippo Population No. of Hippos Day 0 1 2 3 4 5 6 7 8 10 9 Monday Tuesday Wednesday Thursday Friday Car Sales No. of cars sold
Geometry STATISTICS 331 4. David, a meteorologist, discovered the fluctuation in annual air temperature (in °F). He created a line graph that showed the data that was captured. Read the graph and answer the questions. (a) What is the highest recorded temperature? (b) Where do you see a raise 10o F in the graph? (c) Which is the coldest month? (d) Which month recorded 80o F? (e) What is he temperature recorded in the month of December? 5. The graph below shows the number of fish caught in a day. Use the graph to answer the questions. (a) What time were the most fish caught? (b) What time were the fewest fish caught? (c) Did the number of fish caught at 12 PM increase or decrease from 11 AM? (d) How many fish were caught at 9 AM? (e) How many fish were caught at 10 AM? (f) Were more fish caught at 10 A.M or at 11 AM? (g) Were fewer fish caught at 10 A.M or at 11 AM? (h) What is the different in the number of fish caught at 9 AM and the number caught at 12 PM? (i) What is the total number of fish caught? (j) Were there at least 5 fish caught at 8 AM? 6. The following table shows the number of tourists (in thousands) who visited Nepal for entertainment during Poush 2051 to Poush 2055: Year 2051 2052 2053 2054 2055 Number of tourists 51.5 50.5 53.2 59.1 56.4 Represent the above data by a line graph. Month December January February March April May June July September August October November 50 55 60 65 70 75 80 85 95 90 Air Temperature Temperature (°F) 0 7 AM 8 AM 9 AM 10 AM 11 AM 12 PM 1 2 3 4 5 6 7 8 10 9 Fishing Trip Results Time Fish Caught
332 The Leading Mathematics - 7 7. Draw a line graph to represent each set of data in the following tables: (a) The number of children in various families are as shown below: Number of children 1 2 3 4 5 Number of family 5 6 4 3 2 (b) The populations (in hundred thousands) of Nepal according to the censuses taken during 2018 to 2048: Year 2018 2028 2038 2048 Population in hundred thousands 51.5 50.5 53.2 59.1 8. (a) The heights of the well-known high peaks from west to east of Nepal are given below. Draw a line graph by joining each consecutive pair of points by a line-segment. Peaks Dhaulagiri Annaurna Manasalu Sagarmatha Lhotse Makalu Kanchanjunga Height in Km 8172 8078 8156 8848 8501 8470 8598 (b) The length of highways of Nepal are given below. Draw a line graph by joining each consecutive pair of points by a line segment. Highways Tribhuwan Araniko Prithvi BP Mechi Koshi Karnali Length in km 160 113 174 198 268 159 232 9. (a) Draw two line graphs on the same graph paper comparing the pass percentages of the boys and girls in an annual examination as shown in the following table: Class VI VII VIII IX X Boys 75 65 65 70 70 Girls 60 50 45 50 45 (b) Draw two line-graphs on the same graph paper showing a comparison of the expenditures of two families: Items Family Food Clothes Education Transportation Health Rent A 30 10 25 10 10 15 B 35 15 20 10 15 5 ANSWERS Consult with your teacher.
Geometry STATISTICS 333 21.2 Multiple Bar Graph At the end of this topic, the students will be able to: ¾ construct the multiple bar graph from the given data. Learning Objectives Graphical Representations of Data Graphs and diagrams are constructed from the data collected and presented in the form of tables. They provide a pictorial view of the nature of the data. Here, we shall consider the multiple bar diagrams or graph. A bar diagram or graph is a pictorial representation of numerical data by a number of bars or rectangles or pillars. The rectangles are all of the same width. The height or length of each rectangle is proportional to the size of the item it represents. A multiple bar diagram or graph is used for representing two or more data related to each other. In the multiple graphs, the pillars or rectangles with different shadows or colours are used for representing the different variables of the given data. Same multiple bar graph gives the information of the given multiple data. CLASSWORK EXAMPLES Example 1 Construct the multiple graph from the data given below: Classes Students 9 10 11 12 Boys 45 35 30 45 Girls 50 45 35 40 Solution: Constructing the multiple graph from the given data; 0 9 10 20 30 40 50 10 11 12 Classes No. of students Boys Girls Index
334 The Leading Mathematics - 7 Example 2 Study the given multiple diagram and answer the following questions: 0 10 20 30 40 50 6 7 8 Classes No. of Students Boys Girls Index (a) How many boys are there in the class 7? (b) How many total girls are there in the classes 6 and 7? Solution: From the given data, (a) There are 20 boys in the class 7. (b) There are 65 girls in the classes 6 and 7 in total. EXERCISE 21.2 Your mastery depends on practice. Practice like you play. 1. Construct the multiple graphs from the data given below: (a) Number of boys and girls in the classes 5, 6, 7 and 8 of a school; Classes Students 5 6 7 8 Boys 25 20 30 35 Girls 35 25 25 35 (b) Number of boys, girls and teachers in a school in the years 2015 AD, 2016 AD and 2017 AD; Years Students 2017 AD 2016 AD 2015 AD Boys 250 265 270 Girls 240 245 275
Geometry STATISTICS 335 (c) Number of people at the following villages in the different years; Census Village 2058 BS 2068 BS 2078 BS Khokana 3800 4500 4900 Bungamati 4200 5700 6000 Dukuchhap 2000 2500 2700 (d) Number of books of the subjects Mathematics, Science and Computer in the years 2071 BS, 2072 BS, 2073 BS, 2074 BS; Years Books 2071 BS 2072 BS 2073 BS 2074 BS Mathematics 20 24 28 36 Science 16 28 36 40 Computer 8 12 20 30 (e) Number of students who secured the given grades in a school in the SEE of 2073 and 2074; Grading Years A+ A B+ B C+ C 2073 18 12 16 24 20 12 2074 16 20 12 20 16 8 2. Study the given multiple diagram and answer the following questions: 0 6 10 20 30 40 7 8 Classes No. of Students Boys Girls Index (a) How many boys are there in the class 7? (b) How many total girls are there in the classes 6 and 7? (c) How many more boys are there than girls in the class 8?
336 The Leading Mathematics - 7 (d) How many less girls are there than boys in the class 6? (e) Which class has more students? (f) In which class are least girls? 3. Study the given multiple graph and answer the following questions: 0 A 10000 20000 30000 40000 B C Cities No. of people Male Female Index (a) How many people are there in the city A ? (b) What is the total male population in three cities? (c) How many more female population is there than male population in all ? (d) Find the total population in all three cities. 4. Study the given multiple graph and answer the following questions: 0 Sunday 200 400 600 800 Monday Tuesday Days Cost of vegetables Potato Brinjal Index (a) How much potatoes did the shopkeeper sell in total? (b) How much more sells of tomatoes are there than of brinjal? (c) Find the total sales amount. (d) Which item is more sold and by how much? ANSWERS Consult with your teacher.
Geometry STATISTICS 337 Read, Understand, Think and Do 1. A line graph showing the marks obtained in Mathematics by the first five students in a class is as given below. (a) Who got maximum marks in mathematics among five students. (b) What is the differences between lowest and highest marks among five students? (c) In an examination system, less than 35 marks is non graded and 35 to 40 is D grade. How much additional marks should be needed for secured D grade? 2. The number of children in various families are shown below. Number of children 1 2 3 4 5 Number of family 5 6 4 3 2 (a) How many families have 5 children? (b) Draw a line graph. (c) Is the line graph straight? Justify. 3. Class Teacher Ramesh recorded the number of students with boys and girls as follows. Class Students 5 6 7 8 Boys 25 20 30 36 Girls 36 25 25 36 (a) How many students are there in class 8? (b) Draw a multiple bar diagram of the above data. 1st 2nd 3rd 4th 5th 6th 10 20 30 40 50 60 70 Students Marks MIXED PRACTICE–VI
338 The Leading Mathematics - 7 4. A line graph shows the earning by a people in 5 days of a week working in an international organization of Nepal. Mon Tues Wed Thur Fri 100 200 300 400 500 600 700 Days Earning (a) When did he earn maximum amounts in a day? (b) What is the difference between the lowest and the highest earning among 5 days? (c) How much amount does he earn in a week? 5. An electrician charges Rs. 50 plus and Rs. 50 per hour of work he/she does. (a) Complete the table to show the costing. Time (Hr) 1 2 3 4 5 6 Cost (Rs.) (b) Plot the graph to model the cost against time? 1 2 3 4 5 6 7 50 100 150 200 250 300 350 Time (hours) Cost (c) It takes 3 hour 30 minutes to complete the job. How much would the electrician have charged? ANSWERS Consult with your teacher.
Geometry STATISTICS 339 4. A line graph shows the earning by a people in 5 days of a week working in an international organization of Nepal. Mon Tues Wed Thur Fri 100 200 300 400 500 600 700 Days Earning (a) When did he earn maximum amounts in a day? (b) What is the difference between the lowest and the highest earning among 5 days? (c) How much amount does he earn in a week? 5. An electrician charges Rs. 50 plus and Rs. 50 per hour of work he/she does. (a) Complete the table to show the costing. Time (Hr) 1 2 3 4 5 6 Cost (Rs.) (b) Plot the graph to model the cost against time? 1 2 3 4 5 6 7 50 100 150 200 250 300 350 Time (hours) Cost (c) It takes 3 hour 30 minutes to complete the job. How much would the electrician have charged? 1. Observe the given diagram and answer the following questions. Mon Tues Wed Thur Fri 1 2 3 4 5 6 8 7 9 10 Days Number of boxes (a) What represents the given diagram? [1] (b) In which day was the cookies sold maximum ? [1] (c) How many total cookies are sold in all days? [1] 2. The given table shows the number of new admission students from classes 1 to 5. Class 1 2 3 4 5 No. of students 10 15 25 10 20 (a) Construct a line graph from the above data. [2] (b) How many new students admitted from the classes 1 to 5 ? Find it. [1] 3. Observe the given diagram that shows the number of boys and girls from the classes 6 to 10. Class 6 7 8 9 10 Boys 20 25 15 20 25 Girls 15 20 25 15 10 (a) Draw two line graphs on the same graph paper. [2] (b) Find the total number of students from the classes 6 to 10. [1] Attempt all the questions. FM : 15 Time : 35 Min. CONFIDENCE LEVEL TEST VI STATISTICS
340 The Leading Mathematics - 7 4. Study the given multiple graph and answer the following questions: Sunday 0 100 200 300 400 Monday Tuesday Days Cost of Fruits Apple Grape Index (a) How much apples did the shopkeeper sell in total? [1] (b) How much more or less grapes are sold than apples? [2] 5. Observe the given data and answer the following questions. Level Primary Lower secondary Secondary Male Teacher 5 10 15 Female Teacher 7 8 6 (a) What represents the given data? [1] (b) Construct the multiple bar graph from the above data. [2]
Geometry STATISTICS 341 1. Observe the sets A = {factors of 12) and B = {odd numbers less than 10). (a) Write the above sets in the listing method. [1] (b) Find the set A ∩ B. [1] (c) Show the above sets in a Venn diagram. [1] 2 Observe the given data : 51, 65, 49, 72, 73 (a) Write the formula to find the median. [1] (b) Find the mean (x) from the given data [2] 3. Observe the following numbers. 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 (a) List the square numbers. [1] (b) Find the cube root of 27 by using prime factorization. [1] (c) Find the LCM of 27 and 33. [2] (d) The cost of 17 books is Rs. 1071. Find the cost of one book. [1] 4. Deepa gave 20% of her income to her son, 30% of the income to her daughter and remaining amount Rs. 4200 to her husband. (a) Calculate the total income of Deepa. [1] (b) Find the amount received by her son and daughter. [2] (b) How many percent of money did she give her husband? Find it. [1] 5. Raju sold a watch for Rs. 3168 at a loss of 20%. (a) Write the formula to find the discount percentage. [1] (b) Find the cost price of the watch. [2] (c) What is the selling price for the profit 10% ? [2] PRACTICE QUESTION SET Class – 7 Subject: Mathematic F.M. : 50
342 The Leading Mathematics - 7 6. (a) Find the circumference of circle where diameter is 14 cm. [1] (b) Find the are of the given triangle. [2] (c) The total surface area of a cubical box is 216 cm2 . (i) Find the area of its base. [1] (i) Find the length of its side. [1] (ii) Find the volume of the box. [1] 7. (a) If x = 5 and y = 7, find the value of 4x2 + 4xy + y2 2xy . [2] (b) Find the quotient of; 18x4 y2 ÷ 6x2 y2 [1] 8. (a) If x + 1 x = 7, find the value of x2 + 1 x2 [2] (b) Solve : 13x – 8 = 52 + x. [2] 9. (a) Draw the graph of an equation 2x + y = 6. [2] (b) Solve 2x + 2 > 10. [1] 10. A triangle PQR has the size PQ = 5.3 cm, ∠P = 60° and ∠Q = 30°. (a) Construct the triangle from the given information. [2] (b) Can we write PQ2 = PR2 + QR2 ? Write with reason. [2] (c) Find the measure of the remaining angle of the triangle PQR. [1] 11. (a) If 5x° and 4x° are the pair of complementary angles then find each angle. [1] (b) Verify experimentally that the opposite sides of parallelogram are equal to each other. [3] 12. (a) Find the coordinates of the images of the line segments joining of the points A(2, – 3) and B(– 1, 2) under reflection on the x-axis. [2] (b) Plot the following points on the graph and join them respectively. A(5, 3), B(1, 2) and C(4, 2) [3] (c) Write the 3-digits bearing of E30°S [1] Q 20 cm R P 10 cm
Geometry STATISTICS 343 SN Chapter/ Lesson Participation (Attendence + Class work): 4 marks Project and Practical Work 36 marks Terminal Exams: 10 marks (50) Total Marks Percent % Grade Grade point Result Remarks Based on main learning outcomes (20) Records of Project and Practical Work (10) Figure, leveling, Oral (6) First (5) Second (5) Subject: The Leading Mathematics Student's Name : ………………………………………………………. Class : 7 Internal Evaluation Sheet Practical Recorded Form for 50 Marks
344 The Leading Mathematics - 7 SN Chapter/ Lesson Participation (Attendence + Class work): 4 marks Project and Practical Work 36 marks Terminal Exams: 10 marks Total Marks (50) Percent % Grade Grade point Result Remarks Based on main learning outcomes (20) Records of Project and Practical Work (10) Figure, leveling, Oral (6) First (5) Second (5)