Acme Mathematics 7

Acme Mathematics 7 2519. Find the measure of unknown angles.(a)z y 80°x a(b) bdca70°(c)zc a 55°x by(d)cb ad x zy(e)y110° x z30°(f)88°y ax 70°z10. Experimentally verify that: (a) Opposite sides of a rectangle are equal.(b) All sides of a square are equal. (c) Diagonals of a square are equal. (d) Diagonals of a rhombus are not equal.11. Name the figure and write its properties.(a) (b) (c)(d) (e) (f)ADBCPSQRMPNOWZXYABCDMNPO

252 Acme Mathematics 7E Pythagoras Theorem Study the given right angled triangle ABC. Here, ∠B = 90° AB is called the perpendicular and denoted by P. BC is called the base and denoted by b. AC is called the hypotenuse and denoted by h. Let P = 3 cm, b = 4 cm and h = 5 cm, thenHence, we have, p2 + b2 = h2or, (Perpendicular)2 + (base)2 = (hypotenuse)2The sum of the squares on the perpendicular and base of a right angled triangle is equal to the squares on hypotenuse. This statement was investigated by Pythagoras. He is a mathematician and philosopher. He was born in Greek in 582 B.C. So, this statement is known as Pythagoras Theorem.Now study the following figures and complete the table.(a)B 8 cm C10 cm 6 cmA (b)CA 30 cm B50 cm 40 cm(c)C BA13 cm12 cm5 cmFigures AB2 BC2 AC2 AB2 + BC2 Result(a) 36 cm2 64 cm2 100 cm2 100 cm2 AB2 + BC2 = AC2(b) 900 cm2 1600 cm2 2500 cm2 2500 cm2 AB2 + BC2 = AC2(c) 25 cm2 144 cm2 169 cm2 169 cm2 AB2 + BC2 = AC2Hence, squares on the hypotenuse of a right angled triangle is equal to the sum of squares on the two remaining sides. B b ChpAh2 = 52 b = 25 2 = 42 p = 16 2 = 32 = 9B Cb2h2p2A

Acme Mathematics 7 253Remember: Hypotenuse (side opposite to right angle) is the longest side.Perpendicular (adjacent side to base) and base (adjacent side to perpendicular) are not fixed.Science 33 + 42 = 52, 3, 4 and are called Pythagorean triplets.Solved ExamplesExample 1 Find the value of x in the given right angled triangle ABC. Solution : Here, AC is hypotenuse. Perpendicular (AB) = 45 cm, Base (BC) = 28 cm Now, using formula, AC2 = AB2 + BC2or, x2 = (45)2 + (28)2or, x2 = 2025 + 784or, x2 = 2809 ` or, x = 53Hence, the value of x is 53 cm.Example 2 Find the value of x in the given right angled ∆ABC.Solution : Here, AC is hypotenuse. Now, using formula, AC2 = AB2 + BC2or, 172 = 152 + x2or, x2 = 172 – 152or, x2 = 289 – 225or, x2 = 64or, x = 8Hence, The value of x is 8.Classwork1. Write 'True' or False' for the following statements.(a) Longest side of a triangle is its hypotenuse.(b) Side opposite to right angle is called base.(c) Side adjacent to right angle is called perpendicular.(d) 3 cm, 4 cm and 5 cm are the sides of right angled triangle.(e) Pythagoras theorem state that in triangle there are two right angles.B 28 cm Cx 45 cmAThe set of numbers 28, 45 and 53 are called Pythagoras tripletsB x C1715A

254 Acme Mathematics 72. Triangle KLM is right angle at L. Calculate the length of MK using Pythagorus theorem.3. In each of the following, the length of the side perpendicular and base of a right angled triangle are given. Find the length of the hypotenuse.(a) Perpendicular = 1.5 cm and base = 2 cm(b) Perpendicular = 6 cm and base = 2.5 cm(c) Perpendicular = 7 cm and base = 17 cm(d) Perpendicular = 14 cm and base = 48 cm4. State Pythagoras Theorem5. Study the given figure and write the relation between p, b and h. hpbExercise 5.41. Find which of the following are sides of a right angled triangle:(a) 1 cm, 1 cm and 2 cm(b) 5 cm, 12 cm and 13 cm(c) 15 cm, 10 cm and 25 cm(d) 50 cm, 14 cm and 48 cm.2. Find the value of x in the following right angle triangles.(a)x8 cm6 cm(b)15 cmx 8 cm(c)x13 cm 12 cm(d)x 11 cm60 cm (e)65 cmx33 cm (f)x70 cm74 cmMK 12 cm L9 cm

Acme Mathematics 7 255(g)x50 cm40 cm(h)x9 cm12 cm(i)6 cm x10 cm6.5 cm3. Which of the following are the Pythagoras triplets ? (a) 8, 10, 12 (b) 7, 24, 25 (c) 2.5, 6, 6.5 4. Find the values of x and y in the following figures.(a)x6 cm y15 cm10 cm(b)x72 cm65 cmy(c)yx 5 cm13 cm15.8 cm(d)x5 cm 12 cm18 cmy(e)5. A ladder 15 m tall when set against a wall of house, just reaches the window at a height of 9 m. How far is the lower end of the ladder from the base of the wall. 6. Ramnarayan travels 13 km due north by taxi and then 15 km due east by bus. How far is he now ? 7. A tree broke at a point but did not form two pieces. Its top touched the ground at a distance of 10 m from its base. If the point where it broke was at a height of 17 m from the ground, what was the total height of the tree before breaking ? 8. The figure along side is the cylinder. Find the height of the cylinder.19.5 cm18 cm25 cmyx4 cm B CA15 cm

256 Acme Mathematics 7DCOABQA B CFigure-1 Figure-2QA B CQ CA B Figure-1 Figure-2Q CA BEvaluationTime: 72 minutes Full Marks: 301. (a) Write the vertically opposite angle of ∠AOC from given figure. [1](b) Write any two dissimilar property of square and parallelogram. [2] (c) Construct ∠MNO where ON = 5cm, ∠MON = 30oand OM = 6cm. [3] 2. Study the given figure. (a) Adjacent angle of ∠ABQ. [1](b) Make the table as given below and fill in the gaps on the basis of above given figure. Also write the conclusion. [2](c) Construct the Δ ABC with AB = 7.5 cm, ∠ABC = 75o and ∠BAC = 60o. [3]Fig no. ∠ABQ ∠CBQ Result12conclusion:3. (a) Find the complementary angle of 50o. [1](b) Find the value of x. [2](c) Experimental verification that when two straight line are intersected to each other the vertically opposite angle are equal. (Two different figures are necessary.) [3]4. (a) Construct an angle of 45o using scale and compass. [1](b) Find the unknown side from the given right angled triangle. [2] (c) Construct ΔPQR in which PQ = 5cm, ∠RPQ = 45o and ∠PRQ = 75o. [3]5. (a) Write the relation of opposite side of a square. [1](b) Measure all the angles of squares and tabulatethe result in the table given below. [2] Figure No. ∠QAB ∠ABC ∠BCQ ∠AQC Result12Conclusion:(c) Construct the ΔABC with AB = 7cm, ∠CAB = 75o and AC = 6.8cm. [3] QA B Cx° 77°C BA4cm5cm

Acme Mathematics 7 2575.3 Congruent ObjectsA Congruent ObjectsStudy the following figures.(i) (ii) (iii)In the figures (i), (ii) and (iii), The objects are same and sizes are also same. Such objects are called congruent objects. The figures are said to be congruent if they have the same shape and size. ΔABC and ΔDEF are given triangles. Measure the side and angles of given traingle and complete the table. Where, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Similarly, sides AB = DE, BC = EF and AC = DF. We use the symbol“ ≅” to denote congruency. Thus, ΔABC ≅ ΔDEF. Since ∠A = ∠D, ∠A and ∠D are called corresponding angles. In the same a way ∠B and ∠E, ∠C and ∠F are corresponding angles. Similarly,Sides AB and DE are corresponding sides.Sides BC and EF are corresponding sides. Sides AC and DF are corresponding sides.Classwork1. If ΔABC and ΔDEF are congruent triangles, then complete the tableB CAE FDParts of ΔABC Corresponding parts of ΔDEF∠A .........................∠B .........................∠C .........................B CAE FD

258 Acme Mathematics 7Parts of ΔABC Corresponding parts of ΔDEFBC .........................CA .........................AB .........................Are ΔABC and ΔDEF congruent? Give reason.Exercise 5.51. Identify the pairs of congruent figures.(a) (b) (c)(d) (e) (f)2. Measure the side and angles of given traingle ΔABC and ΔDEF then complete the table. B C E FA DParts of ΔABC Corresponding parts of ΔDEF∠A .........................∠B .........................∠C .........................BC .........................CA .........................AB ......................... Are ΔABC and ΔDEF conruent?3. Measure the angles and sides of the given triangles and complete measure.Q R X YPW

Acme Mathematics 7 2595.4 SolidsA. Tetrahedron, Octahedron, Cone and CylinderLook at the given figures carefully.The above figure are figures of the solids. A solid has length, breath and height.Name of some solids are given below.cube cone cuboidcylinder pyramid prismEvery solid has its net, model and skeleton. Parts of solids are Faces, Vertices and Edges.breadthlengthheightIt is Vertex.It is Face.It is Edge.

260 Acme Mathematics 7(a) Net, model and skeleton model of some solids(i) Cube (Regular Hexahedron)It is a net of a cube. It is a skeleton model of cube. It is a model of cube.We can make a skeleton by using drinking straws or wheat straws.A cube has → 6 same square faces→ 8 vertices→ 12 edges. A dice is an example of cube.(ii) TetrahedronTetrahedron is a regular solid. It has 4 faces, 4 vertices and 6 edges.It is a net of tetrahedron.It is skeleton of tetrahedron. It is model of tetrahedron.(iii) OctahedronIt is a net of octahedron It is skeleton of octahedronIt is model of octahedron(iv) CylinderIt is cylinder. It has 2 circles at the 2 ends and curved surface. It has no vertices.

Acme Mathematics 7 261→ It is a net of cylinderDrum, gas cylinder, juice can are examples of cylinder.(v) DodecahedronIt is a net of dodecahedron It is skeleton of dodecahedronIt is model of dodecahedronIt is a regular solid. It has 12 surfaces. Its surface is regular pentagon. It has 20 vertices and 30 edges. (vi) IcosahedronIt is a net of icosahedron It is skeleton of icosahedronIt is model of icosahedronIcosahedron is a regular solid. It has 20 surfaces. Its surface is equilateral triangle. It has 12 vertices and 30 edges.(vii) ConeIce-cream Funnel Birthday capThe above objects are examples of cone. Its base is circular in shape and the surface is curved. It's curved surface meets at a point. It has a vertex.VertexSurfaceBase

262 Acme Mathematics 7With the help of the given figures try to construct the cone.OA BO ABO AB(b) Relation between faces, vertices and edges of solids.Take a cube. Count its vertices, edges and faces. In the figure along side a cube is given.It has 6 faces, 12 edges and 8 vertices.Here, V + F – E= 8 + 6 – 12= 14 – 12= 2Thus, in any regular polyhedra V + F – E = 2 is true.This rule was developed by Euler. So it is also called Euler's formula. This is also called relation between faces, vertices and edges.Classwork1. Fill in the blanks :(a)These figures are .................. triangles.(b)These figures are .................. figures.(c) OA is called ..........................BC is called ..........................AB CO

Acme Mathematics 7 263(d)It is net of .....................(e)It is net of .....................2. If ΔABC and ΔXYZ are congruent, then name the pairs of corresponding parts.3. Draw a net of octahedron.4. Take a tetrahedron and verify the Euler's formula.Exercise 5.61. Write 'True' or 'False' for the following statements.(a) All faces of tetrahedron are equilateral triangle.(b) All sides of tetrahedron are equal.(c) Octahedron has 7 vertices.(d) Cone has no vertex.(e) Cylinder has no faces.2. Identify the shape represented by each of following nets.(a) (b) (c)............................ ............................ ............................(d) (e) (f)............................ ............................ ............................B C X YA Z

264 Acme Mathematics 73. Fill in the blanks. (a) A tetrahedron has ............ vertices and ............. faces. (b) An octahedron has .......... faces and ............ edges. (c) A cone has ......... circular bases and .......... vertex. (d) A cylinder has .......... surfaces ........... bases. 4. Complete the table given below. Regular PolyhedronNumber of Vertices (V)Number of Faces(F)Number of Edges (E)V + F – E ConclusionTetrahedronCubeCuboidHexahedronOctahedron5. A tetrahedron has 4 vertices and 4 faces, find its number of edges.6. Define the following:(a) Cone(b) Cylinder(c) Octahedron7. Name the vertices, edges and faces of the following objects.(a) (b) (c)8. Colour the faces of given figure.(a) (b) (c)AD CE FH GBACDBPS RU VX WQD

Acme Mathematics 7 2651. Study the given figure alongside and fill in the blanks: (a) The line OX is called ................ (b) The line OY is called ................ (c) The point O is called ................ (d) The plan YOX is called ................ 2. In the point A(2, 4), 2 is called .................. and 4 is called .................. 3. From the graph given below, write the coordinates of :BA DCOYX(a) A (b) D(c) B (d) C4. Plot the following points on the graph. (a) P(4, 0) (b) Q(6, 2)(c) A(7, 3) (d) B(5, 5)OY'YX' XWarm Up Test5.5 Co-ordinates

266 Acme Mathematics 7A Introduction to x-axis and y-axisCo-ordinates are used for locating and communicating a position. The location of the point P in the figure alongside is 2 units across and 3 units upward from O.Thus, the position of the point P can be represented by a pair of numbers (2, 3). This P (2, 3) is known as coordinates. In general the coordinates of any point is (x, y).Where is the point Q located? Discuss. (a) Quadrants On a graph paper, draw the lines XOX' and YOY' perpendicular to each other at O. The line XOX' and YOY' through O are called axes. The point of intersection O is called the origin. The horizontal line XOX' is called the x-axis and vertical line YOY' is called the y-axis. The x-axis and y-axis divide the graph into four regions which are called the quadrants. These quadrants are as follows: (i) First quadrant: The region XOY is the first quadrant. (ii) Second quadrant:The region YOX' is the second quadrant. (iii) Third quadrant:The region X'OY' is the third quadrant. (iv) Fourth quadrant: The region Y'OX is the fourth quadrant. (b) Sign in the quadrants (i) The region XOY is the first quadrant. In this quadrant value of x is positive and y is positive. (ii) The region YOX' is the second quadrant. In this quadrant value of x is negative and y is positive. (iii) The region X'OY' is the third quadrant. In this quadrant value of x is negative and y is negative. (iv) The region Y'OX is the fourth quadrant. In this quadrant value of x is positive and y is negativeO XYPQOSecond QuadrantThird QuadrantFirst QuadrantFourth QuadrantX' XY'Y(–, +)(–, –)(+, +)(+, –)O X' XY'Y

Acme Mathematics 7 267B Reading the co-ordinates of the pointsLet us observe the following points in the graph paper. 1. We write the x co-ordinate first and then y co-ordinate in the order pairs. 2. The coordinates of the origin is (0, 0). 3. The point P lies in the first quadrant. The coordinates of point P is (4, 3). Why ? 4. The point Q lies in the second quadrant. The coordinates of point Q is ( – 3, 2). 5. The point R lies in the third quadrant. The coordinates of point R is ( – 2, – 3). 6. The point S lies in the fourth quadrant. The coordinates of point S is (3, – 1) In the order pair (4, 3), the numbers 4 and 3 are called the coordinates. 4 is x-coordinate and 3 is y - coordinate. The x- coordinate is called the abscissa and the y-coordinate is called the ordinate of the point. The co-ordinates of any point on the x-axis is (x, 0). The co-ordinates of any point on the y-axis is (0, y).C Plotting the points on the graphWhile plotting the points, we first move along x-axis and then along y-axis. To the right of the origin, the value of x is positive and to the left it is negative. Similarly, the value of y is positive upwards the origin and negative, downwards the origin.Classwork1. Write the co-ordinates of the following points from the given graphs: (a) E (b) D (c) C (d) B (e) A 2. Write the quadrants for the following points: (a) (2, 3) (b) (– 4, 7) (c) (x, – y) (d) (– 7, – 9)(e) (– 2, 13) (f) (14, – 7) (g) (– x, – y) (h) (– 17, 9)Q(–3, 2) P(4, 3)S(3, –1) R(–2, –3)O X' XY'YEDCABO 1153726489102 3 4 5 6 7 8 9 10 X' XY'Y

268 Acme Mathematics 73. Plot the following points on the graph and join the points: (a) A(2, 3), B(6, 7) (b) A(4, 5), B(6, 0)(c) A(3, 2), B(0, 7) (d) A(5, 6), B(–2, 3) 4. Fill in the blanks: (a) The region X'OY is called ................ (b) The region XOY' is called ................ (c) The coordinates of origin is ..............5. Write true or false for the following statements:(a) The coordinates of any point on the x-axis is (x, x)(b) The y coordinates is also called the ordinates of the point.Exercise 5.71. Write the co-ordinates of the vertices of ΔABC from each of the following graphs: (a)OBACX' XY'Y (b)OCABX' XY'Y (c)OBCAX' XY'Y(d)OACBX' XY'Y (e)OAB CX' XY'Y (f)OBCAX' XY'Y2. Plot the following points on the graph and join the points: (a) A(–2, 3), B(6, 0) (b) A(–2, –3), B(4, 5)(c) A(4, –5), B(6, 3) (d) A(–3, –2), B(0, 5)(e) A(6, 7), B(–2, –3)OY'YX' X

Acme Mathematics 7 2693. Plot the following points on the graph and join them in order. What does the shape represent? (a) A(2, 3), B(4, 5), C(6, 2), A(2, 3) (b) A(3, 2), B(6, 3), C(4, 6), A(3, 2) (c) A(1, 1), B(1, 3), C(4, 3), D(4, 1), A(1, 1) (d) A(2, 2), B(2, 4), C(5, 4), D(5, 2), A(2, 2) (e) A(3, 1), B(–4, 4), C(–1, 7), D(6, –2), A(3, 1) (f) A(2, 3), B(–2, 3), C(–2, –3), D(2, – 3), A(2, 3) (g) A(–3, 2), B(1, 2), C(1, –1), D(–3, – 1), A(–3, 2) 4. Following points are the three continuous vertices of a rectangle. Find the fourth vertex D, graphically:(a) A (–1, 2), B (2, 2), C (2, 0) (b) A (–2, –1), B (1, –1), C (1, 3) (c) A (–3, – 2), B (0, – 2), C (0, 2) (d) A (1, 1), B (4, 1), C (4, 3)(e) A (1, – 2), B (6, –2), C(6, – 1) (f) A (1, 2), B (4, 1), C (5, 3) (g) A (2, 1), B (5, 0), C (6, 2) (h) A (–2, 0), B (1, – 1), C (2, 1) 5. Following points are the vertices of a rectangle. Plot the points on graph and join in order. (a) A(2, 2), B(6, 2), C (6, 4), D (2, 4) (b) A (1, 2), B (5, 2), C (5, 4), D (1, 4) (c) A(0, 2), B(4, 2), C (4, 4), D (0, 4) (d) A (0, 1), B (4, 1), C (4, 3), D (0, 3) (e) A(2, 1), B(6, 1), C (6, 3), D (2, 3) 6. Plot the points P (–1, 3), Q (2, 3), R (–1, –1) and S (– 4, –1). If PR and QS meet each other at M, find the coordinates of M. 7. Plot the points A (2, 2), B (6, 2), C (6, 4) and D (2, 4). If AC and BD meet each other at M, find the co-ordinates of M. 8. Plot the points A (1, 2), B (5, 2), C (5, 4) and D (1, 4). If AC and BD meet each other at M, find the coordinates of M. 9. Plot the points A (0, 2), B (4, 2), C (4, 4) and D (0, 4). If AC and BD meet each other at M, find the coordinates of M. 10. Plot the following points on a sheet of graph paper: A (5, 5), B (4, 4), C (2, 2), O (0, 0), D(–2, –2), E (– 4, – 4). Join all of these points in order. What does the shape represent?

270 Acme Mathematics 75.6 Distance between two points(a) Vertical lineStudy the graph along side. (a) How long is line AB ?(b) How long is line CD ?Now, count the steps from the point A.Point B is 3 units above from A. So, the line AB is 3 units long.Similarly, the line CD is 4 units long.The vertical distance is called ‘rise’ part of a line.We denote the point A(2, 1) as (x1, y1) and B(2, 4) as (x2, y2).Rise = difference between y-values = y2– y1 = 4 – 1 = 3 units.(b) Horizontal lineStudy the graph along side. (a) How long is line AB ?(b) How long is line CD ?Now, count the steps from the point A.Point B is 3 units right to A. So, the line AB is 3 units long.Similarly, the line CD is 4 units long.The horizontal distance is called ‘run’ part of a line.We denote the point A(1,3) as (x1, y1) and B(4, 3) as (x2,y2).Run = difference between x-values = x2 – x1 = 4 – 1 = 3 units.(c) Slant lineLet, O(0, 0) and B(3, 4) be two points on the graph. Join the points O and B. OB is a slant line. Draw BA perpendicular to OX.YY'O X' XB(2, 4)A(2, 1)C(–3, – 1)D(–3, 3)YY'O X' XA(1, 3) B(4, 3)D(–7, 1) C(–3, 1)YY'O X' X (0, 0) A(3, 0)B(3, 4)

Acme Mathematics 7 271Now, ‘Run’ for line OB = OA = x2 – x1Where, O(x1, y1) = (0, 0) and A(x2, y2) = (3, 0).OA = 3 – 0 = 3‘Rise’ for line OB = AB = y2 – y1Where, A(x1, y1) = (3, 0) and B(x2, y2) = (3, 4).AB = 4 – 0 = 4Since, ∆OAB is a right angled triangle,right angled at A, then,OB2 = OA2 + AB2 (Pythagoras relation)= (run)2 + (rise)2= (x2 – x1)2 + (y2 – y1)2= 32 + 42= 9 + 16or, OB = 25or, OB = 5 units.Note: If (x1, y1) and (x2, y2) are two points and 'd', the distance between the points then we have, distance (d) = (run)2 + (raise)2∴ d = (x2 – x1)2 + (y2 – y1)2. It is known as distance formula.Solved ExamplesExample 1 Study the given graph. How long is line AB ?Solution: Run for line AB = 3 units Rise for line OB = 4 units Now,AB2 = (run)2 + (rise)2= 32 + 42= 9 + 16or, AB = 25or, AB = 5 units.Hence, the line AB is 5 units long.A(x1, y1)B(x2, y2)dYY'A OBX' X

272 Acme Mathematics 7Example 2 Find the distance between the points A and B.Solution: Here, coordinates of A = (–1, 1)coordinates of B = (3, 3)Let, (x1, y1) = (–1, 1) and(x2, y2) = (3, 3)d = distance between A and B.Now, by using distance formulad = (x2 – x1)2 + (y2 – y1)2= (3 – (–1))2 + (3 – 1)2= 16 + 4= 20 = 2 5 units Hence, distance between the points A(–1, 1) and B(3, 3) is 2 5 units.Example 3 If P(0, 0), Q(2, 2) and R(6, 7) are three points, (i) Find the distance PQ. (ii) Find the distance QR.Solution: (i) Here, Let P(0, 0) = (x1, y1) and Q(2, 2) = (x2, y2)Using distance formula,PQ = (x2 – x1)2 + (y2 – y1)2= (2 – 0)2 + (2 – 0)2= 22 + 22= 4 + 4 = 8 ∴ PQ = 2 2 units ............(i)(ii) Here, Let Q(2, 2) = (x1, y1) and R(6, 7) = (x2, y2)Using distance formula:QR = (x2 – x1)2 + (y2 – y1)2= (6 – 2)2 + (7 – 2)2= 42 + 52= 16 + 25 = 41 ∴ QR = 41 units YY'OABX' XP(0, 0)Q(2, 2)R(6, 7)

Acme Mathematics 7 273Classwork1. Find the length of the given line segments. (a) (b)2. Draw a rectangle on the graph and name it. And color the rectangle.3. The points A(–2, 1), B(0, 2), C(4, 2) and D(2, 1) are the vertices of a parallelogram ABCD. Calculate the length of the diagonals AC and BD.Exercise 5.81. Find the distance between the given points .(a) A(0, 0) and B(3, 2) (b) B(0, 2) and C(–2, 2)(c) C(–2, 0) and D(0, 1) (d) D(1, 1) and E(2, 2)2. Find the distance between the given points .(a) A(4, 1) and B(3, 2) (b) B(3, 2) and C(–2, 2)(c) C(–2, 2) and D(–1, 1) (d) D(–1, –1) and E(2, 4)3. Three points are A(1, 0), B(0, 7) and C(4, –2). Find the distance between the points,(a) A and B (b) B and C (c) C and A4. Vertices of the triangle PQR are the points P(1, 1), Q(0, 0) and R(5, –1) .Calculate the length of the sides PQ, QR and PR.5. The points A(–2, 1), B(0, 2), C(4, 2) and D(2, 1) are the vertices of a parallelogram ABCD. Calculate the length of the sides AB, BC, CD and AD.YY'OBDFECQPAX' XYY'OBDE FCP QAX' XQRP

274 Acme Mathematics 75.7 Symmetry and TessellationsA Symmetry1. Line of SymmetryLook at the following figures.mmmIn the above figures. Line 'm' is called the line of symmetry. These figures are called symmetrical figures. The lines which divides the figures into two equal halves is called the axis or line of symmetry.2. Rotational SymmetryWhen a shape can be original position by a rotation of less than 3600, it is called rotational symmetry.Study the given figure carefully.In our day to day lives, we see that many objects rotates. Rotation is the circular movement of an object about a point.Line of symmetry Two identical partsObject

Acme Mathematics 7 275Rotation can be clock-wise or anticlock wise. When we tight the screw it is clock wise rotation (in the direction of rotation of hands of clock) and when we open the screw it is anti-clock wise rotation.Centre of rotationThe rotation is always about a fixed point. The fixed point is called centre of rotation.Centre of rotationAngle of rotationThe angle of rotation is an angle about which an object is rotated.(i)(ii)(iii)120°120°120° angle of rotation is 120°Original position 90° rotation 180° rotation Original positionOriginal position 90° rotation 180° rotation Original position 90° rotation 180° rotation

276 Acme Mathematics 7In the above first and third figures angle of rotation is 90° and clockwise direction. In the second figure angle of rotation is 120° and direction is clock wise. The rotation of fan is anti-clock wise.Order of rotationThe order of rotation is the number of times a figure takes its original position during a complete (360°) rotation.For examples:Original position 90° rotation + 180° rotation + 270° rotation + 360° rotation (original position)The figure turns into its original position after 360° rotation so it has 1 order of rotational symmetry.It has 2 order of rotational symmetry.It has 3 order of rotational symmetry.Look at the following figures carefullyl1ml2ml1Heart Butterfly Isosceles triangleThis figure fitted onto (completely) the original shape after the rotation of 360° it has 1 order of rotational symmetry.

Acme Mathematics 7 277The following figures have 2 order of rotational symmetry:l2l1Multiplication signl2l1Rectanglel3l2l1Equilateral triangleEquilateral triangle has 3 order of rotational symmetryl4l2l3l1SquareSquare has 4 order of rotational symmetryl1l2l3l5l6A circle has infinite order of rotational symmetry.Parallelogram and scalene triangle has no rotational symmetry3. Point symmetry Point symmetry means a shape looks the same after being rotated 1800 around a central point, where every point on one side has a matching point the same distance away on the opposite side. It creates an upside down identical image.Many playing cards have point symmetry, so that they look the same from the top or bottom.A BB AO O1800

278 Acme Mathematics 7Classwork1. Fill in the blanks.(a) A circle has ........... lines of symmetry .(b) An equilateral triangle has ........... lines of symmetry.(c) A scalene triangle has ........... lines of symmetry.(d) A square has ........... lines of symmetry.(e) A rectangle has ........... lines of symmetry.(f) The order of rotational symmetry of a square is ........... .(g) The angle of rotation of an equilateral triangle is ........... .(h) The order of rotation of a circle is ........... .(i) The order of rotation of a line segment is ........... .(j) The order of rotation of the letter H is ........... .2. Write whether the following statements are true or false.(a) The letter N of English alphabet has no line of symmetry. (b) The letter H of English alphabet has one line of symmetry.(c) The letter A of English alphabet has two lines of symmetry.(d) A parallelogram has no line of symmetry.(e) The diameter of a circle is the line of symmetry of the circle.(f) A square is symmetrical about its diagonal.(g) Order of rotation is 360°.(h) Order of rotation of 180°.(i) Order of rotation of a circle is infinite.Exercise 5.91. Draw the line of symmetry for the following figures, if possible.(a) (b) (c) (d)(e) (f) (g) (h)

Acme Mathematics 7 2792. Draw the order of rotational symmetry for the following figures.(a) (b) (c) (d)(e) (f) (g) (h)B TessellationTessellation is covering of the surface with regular congruent geometrical shapes in a repeating without any gaps. It is used in the surface, walls, floor or carpets to make the area more attractive. We can make attractive designs by drawing many triangles, squares, pentagons and hexagons. Look at the following designs.A tessellation of triangles A tessellation of square A tessellation of hexagonsIn all the figures a tessellation is created when a shape is repeated over again covering a surface without any gaps. It is regular tessellation.The following tessellations are the semi tessellationTessellation of hexagon and dodecagonTessellation of octagon and squareTessellation of triangle and parallelogram

280 Acme Mathematics 7The tessellations given below are the irregular tessellation.Exercise 5.101. Copy the given tessellation and complete it in A4 size paper.2. Using the regular polygons given below make different tessellations taking at list two polygons at a time.(a) (b) (c) (d)3. Copy the given tessellation and complete it.(a) (b) (c)(d) (e) (f)

Acme Mathematics 7 2815.8 TransformationIntroductionLook at the following pictures. (a) (b)(c)In the pictures, object is moved from one place to another place by 3 different ways. This process of shifting the object from one place to another is called transformation. Some transformations are given below.A. TranslationIn the figure ΔABC is moved to ΔA'B'C'. ΔABC is translated to ΔA'B'C'. ΔA'B'C' is called the image of ∆ABC. When an object is moved in a straight line for a certain distance it is called translation.Translation on the gridSuppose that Rajiv pushes a triangle A, 4 units to the right. Triangle moves the place A to the place B. Again if he pushes the triangle A, 5 units up then triangle moves from the place B to place C.Triangles B and C are called the images of triangle A.CAC'A'B'BA BC4 units5 units

282 Acme Mathematics 7Movement on the grid:(a) Right - up ......................(b) Right - down .................(c) Left - up ........................(d) Left - down ...................Solved ExamplesExample 1 Draw the ΔABC in the new position after the translation of 4 cm along with BC.Solution: Steps:(i) Draw lines through points A, B and C and parallel to the BC.(ii) Cut the lines AA' = 4 cm, BB' = 4 cm and CC' = 4 cm.(iii) Join the points A', B' and C'.4 cmB CAB' C'A'ΔA'B'C' is the translation image of ΔABC, with 4cm along with BC.Example 2 Shift the quadrilateral ABCD, 6 unit right and 7 units down and find image quadrilateral A'B'C'D'.Solution: In the grid quadrilateral A'B'C'D' is the image of quadrilateral ABCD after translation.CC'DD'AA'BB'6 units7 unitsStarting point is the any given point of the objectB CACDAB

Acme Mathematics 7 283Exercise 5.111. Translate the given figures 5 cm each in the gives directions.(a) A Right from the point A.(b)ABRight and parallel to AB.(c)BACIn the direction of BC.(d)A BCIn the direction of AC.2. Find the image of given figures in any direction with 7 cm displacement.(a)B CA (b)Q RP (c)EDF(d) ADBC(e) SQP R(f) GH FD E3. In the square grid draw triangle ABC, where coordinates of A, B and C are (4, 2), (5, 4) and (2, 5) respectively. Now translate the ΔABC in the given conditions and write the name of image triangle.(a) 4 units right and 6 units up(b) 6 units left and 7 units up(c) 8 units left and 4 units down (d) 5 unit right and 5 units down

284 Acme Mathematics 7B. ReflectionIntroductionFig (i) Fig (ii) Fig (iii)In the figure (i), you can see a cat and its image in the mirror. In the figure (ii), you can see a child and its image in the mirror.In the figure (iii), you can see a dog and its image in the lake. All these figure gives the idea about reflection.Object triangle A PLine 'l' is called mirror lineImage triangleC C'QB B'RA'lMirror line is also called axis of reflection. When an object is moved by a mirror line to form its image is called reflection. Now,(1) Measure the length of AP and PA':Measure the length of CQ and QC': Measure the length of BR and RB':(2) Measure the size of ∠P, ∠Q, and ∠R. What is your conclusion?™ The object and its image are at the equal distance from the mirror line.™ AA', BB' and CC' are perpendicular to mirror line at P, Q and R respectively. ™ Object and image are of the same size (congruent).

Acme Mathematics 7 285Reflection on the gridDraw ΔABC on the grid. Let 'l' be the mirror line.Now, point A is 3 units left from the mirror line so its image point A' is 3 units right from the mirror line.Similarly,B' is the image of the point B. C' is the image of the point C.A', B' and C' are joined.Thus, ΔA'B'C' is an image of ΔABC, after reflection.Solved ExamplesExample 1 Find the image of Δ ABC after reflection on the line 'm' as mirror line.Solution: Steps: Draw line AA', BB' and CC' perpendicular to mirror line. Mark the point A' such that PA =PA'.Mark the B' such that QB = QB'Mark the point C' such that RC = RC' Join the A', B' and C'.RQPmB'A'C' CBA∆A'B'C' is an image of ∆ABC after reflection.Classwork1. Copy and draw the image of the following figures. Dotted lines are the line of reflections.(a) AB Cm(b)BACD m(c) PQ SRmABC C'B'A'Mirror line (l)mCBA

286 Acme Mathematics 72. Copy and draw the image of the following figures.(a)AC Axis of reflectionmB(b)Axis of reflectionAB Cx(c) Axis of reflectionB FDC GrEAExercise 5.121. Copy and draw the image of the given points or lines on the line 'm' as axis of reflection (mirror line)(a)Am (b) Bm(c) C m(d) BAm (e)XYm(f) ABm2. Try your mirror with following figures and complete it.(a) m (b) m (c) m(d)m(e) m (f)m

Acme Mathematics 7 287C. Reflection using co-ordinates(i) Reflection about x-axis. (ii) Reflection bout y-axis. (i) Reflection about x-axis Let, A(2, 3) be the point and x-axis (X'X) is the axis of reflection. Now, Draw AR⊥X'X and produce AR to A' so, that AR = RA', where coordinates of A' is (2, –3). In short, A(2, 3) Reflection about x-axis A'(2, – 3)Thus, while reflecting a point in x-axis, the x-coordinate is unchanged and the y-coordinate has the opposite sign. In general, we have, A(x, y) Reflection about x-axis A'(x, – y) . Example 1 The vertices of ΔABC are A(2, 3), B(4, 5) and C(6, 2). Reflect it in x-axis . Solution: Here, in the figure alongside ΔA'B'C' is an image of ΔABC. (ii) Reflection about y-axis In the adjoining figure let, A(– 2, 3) be the given point and y-axis (YY') is the axis of reflection. Now draw AR perpendicular to YY' and produce AR to A' such that AR = RA', where coordinates of A' is (2, 3). In short, A(–2, 3) Reflection about y-axis A'(2, 3) Thus, while reflecting, a point on y-axis, the y-coordinate is unchanged and the x-coordinate has the opposite sign. In general, we have, A(x, y) Reflection about y-axis A'(– x, y) O RA(2, 3)A'(2, –3)YY'X' XOA(2,3)B(4,5)C(6,2)A'(2,–3)C'(6,–2)B'(4,–5)YY'X' XORA(–2, 3) A'(2,3)YY'X' X

288 Acme Mathematics 7Solved ExampleExample 1 The vertices of triangle ABC are A(1, 0), B(3, 2), and C(1, 4) . Reflect it about y-axis and write down the coordinates of the vertices of image triangle A'B'C'. Solution: Here, in the figure alongside triangle ABC is reflected about y-axis. Triangle A'B'C' is an image of triangle ABC.From the graph coordinates of vertices of triangle A'B'C' are A'(– 1, 0), B'(– 3, 2) and C'(– 1, 4).Exercise 5.131. Reflect the following objects by taking m as axis of reflection.(a) BAm(b) AB C m(c) A BCm(d)m(e) m (f)m2. Reflect the point A(4, 5) on the graph when the axis of reflection is given below: (a) x-axis (b) y-axis 3. The line AB joining the points A(3, –3) and B(15, 5) is reflected on the given axis: (a) x-axis (b) y-axis Find the images of the line AB. 4. The vertices of ΔTMP are T(5, 8), M(5, 6) and P(7, 7) . Plot the ΔTMP on the graph and find the coordinates of the image of the vertices of ΔTMP when reflected on the following reflection axis: (a) x-axis (b) y-axisOC'(–1,4) C(1,4)A'(–1,0)B'(–3 ,2) B(3 ,2)A(1,0)YY'X' X

Acme Mathematics 7 289EvaluationTime: 58 minutes Full Marks: 241. P(–3, 3), Q(4, 3) and R(1, 1) are the vertices of triangle PQR. (a) Plot the vertices of triangle PQR on graph. [2](b) Reflect it about x-axis. Also, write the co-ordinates of its image. [2](c) Find the length of PP'. [1]2. Observe the given graph as given below. [1 small box = 1 unit](a) Write the coordinates of the points P,Q and R . [2](b) Reflect the points P and Q in x- axis and thenwrite the coordinates of their images. [2](c) Why coordinates of the point R and its image R' is same? Write reason. [1]3. Observe the given figure.(a) What is the line ‘O’ called? [1] (b) Draw the image ∆ RKB through the line ‘O’. [2](c) Are the image traingle is equal in size? [1]4. Observe the given gigure along side. (a) What is the line 'm' called ? Draw the image of the line SR . [2] (b) Write the relation between the lines RR' and SS' with the line 'm'. [2](c) Are the lines RS and R'S' equal ? Write reason also. [1]5. Observe the given gigure along side.(a) What is the line ' ' called ? Draw the images of the figure ABCD . [2] (b) Write the relation between the lines AA' ,BB', DD' and CC'. [2](c) Are the lines DC and D'C' equal ? Write reason also. [1]PQ RPOX' XYY'QRmRSA BD CRB KO

290 Acme Mathematics 75.9 Bearing and Scale DrawingA BearingLook at the given compass. It shows the directions. North (N), South (S), East (E) and West (W). WE is like x-axis. NS is like y-axis. More Direction on the Compass NE represents the direction North-East. The direction NE lies exactly in between North and East. The angle between O and E is 90°. The angle between O and NE is 45°. The angle between O and S is 180°. In the bearing system we take NS (North-South) as the base line to measure the angle. O is the starting point for the bearing. Bearing is measured in degree measure in clockwise direction. We write bearing in 3-digit figure. The compass bearing is also called three figure bearing. Hence, the direction of E from O is 090°. It is called the bearing of E from O.B Scale DrawingConsider the figures given below.In the above figures, size of elephant and ant both are not real. These are the drawing only. Here, elephant is reduced to very small size. Figure of ant is enlarged to big size. A drawing that shows a real object either in large size or in small size is scale drawing. Scale indicates the ratio between the size of real object and its image. For example: the N SOSSW SEW ENW NEN

Acme Mathematics 7 291height of tower 100 m can be represented by the given figure along side.0 10Scale 1 cm : 10 m5 15 20 10 cmSolved ExampleExample 1 Write the bearing of the points A, B, C and D Solution: (i) Bearing of A is 045° (ii) Bearing of B is 135° (iii) Bearing of C is 225° (iv) Bearing of D is 315°Example 2 Convert the following measures to the three figure bearing.(a)OS A30°W EN (b)OSW ENB 70°Solution: (a) The position of A is S30°W. So, its 3-figure bearing = 180° + 030° = 210° (b) The position of B is N70°W. So its 3-figure bearing = 360° – 070° = 290° Example 3 The figure given alongside is the map of study room of Mukunda. Find its scale. Solution: Here, Actual length of DC is 4 m. Actual breadth of BC is 3 m But in the figure (map) its length DC = 4 cm and breadth BC = 3 cm Now, Scale = Length in figureActual length = 4 cm4 m = 1 cm : 1 mHence, the scale is 1 cm : 1 mOSC BW ED AND 4 m3 mACB

292 Acme Mathematics 7Classwork1. Write down the bearing of the places A, B and C.(a)70°NA O(b) (c)CN50°O2. Measure the angles and convert to the 3-figure bearing for the points A, B, C, D and R. 3. Draw the figure to show the following compass bearing (a) N 40° E (b) S 40° E (c) S 40° W (d) N 40° W Exercise 5.141. Convert to the 3-figure bearing: (a) N 88° E (b) S 56° W (c) S 70° E (d) N 15° W 2. Taking scale 1mm: 1m, find the actual length and breadth of the following objects. (a) (b) (c)3. A rectangular ground is 200 m long and 150 m broad. Using 1 mm : 10 m as scale, draw the figure of the ground.4. Find the actual height of the man whose photo is given alongside.Scale = 1 mm : 3 cmO 120°NBOSCDBW ER AN

Acme Mathematics 7 2935. Study the map of Bagmati Pardes and answer the following questions.DadeldhuraHumlaSalyanPokharaButwal KathmanduBirgungNamche BazarBiratnagarRolpaN100 km200 km[Take Kathmandu as measuring point](a) Write the bearing of Pokhara. (b) Write the bearing of Biratnagar. (c) Write the bearing of Humla. (d) Write the bearing of Salyan. 6. Study the floor-plan map of a house and answer the following questions. [Scale 1 cm = 2 m] (a) Find the length of bedroom – 1. (b) Find the length of kitchen. (c) Find length and breadth of bath room. (d) Find length and breadth of bedroom – 2. (e) Find length and breadth of the house.1 Passage 2Kitchen BarandaBedroom BedroomBathroom

294 Acme Mathematics 71. (a) What is vertically opposite angle?(b) Draw an angle 120° by using compass.(c) From the figure, find the value of x and y.x°2x°AB CD5y° 4y°A O BC2. In the given figure ∠ MNO = 90° then write.(a) Adjacent angle of ∠ MNP.(b) Write the complementary angles of ∠ PNO.(c) Write the supplementary angles of 120°3. (a) Construct the angle 75° using compass.(b) Draw the angle 60° using protractor then construct equal angle using compass.(c) Find the value of 'a'.4. (a) Find the value of x, y and z.(b) Experimentally verify that: \"The sum of adjacent angle in a straight line is 180°.\"5. (a) What is the measure of angle formed by a revolving a line in a complete rotation at a point?(b) In the given figure, find the measure of 'a' and 'b'.(c) Construct a triangle ABC in which AB = 5.6 cm, AC = 4.1 cm and ∠A = 45°.MN OPa°55°150°b°AO CBa45°AByzxDC70°Mixed Exercise

Acme Mathematics 7 2956. (a) Define complementary angles.(b) Experimentally verify that opposite sides of parallelogram are equal. (two figures of different size of parallelogram are required)7. Do the following.(a) Find the size of ∠COB from the adjoining figure.(b) Draw 150° angle by using compass.8. Line segment and angles are to be drawn.(a) Draw a line segment AB = 6 cm.(b) Draw an angle of 75° using compass.(c) Bisect the angle drawn with compass and measure both angles.9. Draw an angle of 110° using proctractor.(a) Copy the angle in the side using compass.(b) Write the supplementary angle of 110°. (c) If two complememtary angles are in the ratio of 4:5, find the angles.10. Observe the following figure and answer the following questions.(a) Define vertically opposite angles.(b) Find values of a, b and c.11. (a) Is every rectangle a parallelogram? Give reason.(b) In the figure, a ladder rest against a vertical wall at a height of 15 ft. If the foot of the ladder is at a distance of 20 ft. from the wall on the ground, what is the length of the ladder? Find it.(c) Draw the net of a tetrahedron.12. (a) A cube has 6 faces and 8 vertices then find its number of edges. (Using Eulars' formula)(b) Construct a triangle ABC, having side AB = 5 cm, ∠ABC = 60° and side BC = 4.7 cm.(c) Construct ∆LMN in which LM = 6 cm, ∠L = 60° and ∠M = 45°.13. Observe the given solid.(a) Write down the number of faces, edges and vertices of given solid.(b) Test the Euler's formula for above solid.A BC2x x45°2b – 9ca15 ft20 ftO

296 Acme Mathematics 714. (a) The vertices of triangle ABC are A(2, 5), B(–3, 1) and C(4, 0). Reflect the ∆ABC about y-axis then find the coordinates of image ∆A'B'C'. State whether the ∆ABC and ∆A'B'C' congruent or not.(b) If the bearing of Sujan's school from his home is 135°, what is the bearing of his home from the school? Find it. 15. (a) Write the quadrants for the following points. P(–1, 2), X(3, 5), Y(3, – 5)(b) Find the distance between the given points A(4, 1) and B(3, 2).(c) What type of triangle is formed by joining the points P, X and Y? Give reason. 16. (a) In Pythagoras theorem which of the following is true?(i) h2 + b2 = p2 (ii) b2 – h2 = p2(iii) p2 + b2 = h2 (iv) p2 – b2 = h2(b) Find out the missing side of ∆ABC.B 24 cm BA7 cm17. (a) In which quadrant do the points A(–3, – 9) and B(4, – 6) lie?(b) The vertices of ∆ABC are A(– 1, 2), B(4, 1) and C(1, 5) respectively. Write the coordinates of verices of image of ∆ABC obtained on reflecting it in x-axis. Also, draw the image and object on the graph paper.18. A ladder rest against a wall of height 8 ft. 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 ladder makes 20° with the ground, which angle does it make with the wall? Find it.19. (a) Make a figure with one line of symmetry.(b) Write down the bearing of the place A.(c) Construct an equilateral triangle ABC with each side 6 cm. 6 ft8 ft70°NOAB

Acme Mathematics 7 29720. In the given figure straight lines PQ and RS are intersecting at the point O.(a) Write the relationship between vertically opposite angles ∠POS and ∠ROQ.(b) Measure given vertically opposite angle ∠POS and ∠ROQ and tabulate the result in the table given below.Figure No. ∠POS ∠ROQ Result12Conclusion:(c) Construct the 70° and 105° angles by using the compass.21. (a) Construct the ∆ABC with AB = 7 cm, ∠ABC = 60° and ∠BAC = 120°(b) Write the relation of opposite side of a square.22. (a) From the given pairs of congruent triangles, find the value of x and y.5 cm40°60°4 cmB CAyx40°60°zE FD7 cm(b) Write any two difference between tetrahedron and octahedron.(c) Complete the given figure where the dotted line represents the line of symmetry and find out the axes of symmetry.23. (a) Define congruent figures.(b) The vertices of ∆ABC are A(2, 2), B(5, 9) and C(9, 1). Draw ∆ABC in a graph and reflect it about x-axis.(c) What is the complementary angle of 40°?24. (a) Write a difference between dodecahedron and octahedron.(b) In a right angled triangle, if the longest side is 13 cm and base is 5 cm. Find the perpendicular of that triangle.(c) Construct an angle of 90° using compass and bisect it.RO QSPRQOSPFig (i) Fig (ii)

298 Acme Mathematics 725. In the given figure straight lines AB and CD are intersecting at the point E.A BDECACEBD Fig. 1 Fig. 2(a) Write the relationship between adjacent angles AEC and BEC.(b) Make the table as given below and fill in the gaps on the basis of above given figure. Also write the conclusion.Fig no. ∠AEC ∠BED Result12Conclusion:(c) Construct the Δ ABC with AB = 7.5 cm, ∠CAB = 75° and ∠ACB = 75°.26. In the figures, opposite sides are parallel.Fig (i)SPRQFig (ii)P SQ R(a) Write the relation between the angles QPS and QRS.(b) Make a table as given below and fill in the table on the basis of fig (i) and fig (ii). Also write the conclusion.Fig. PQ QR SR PS Conclusion12(c) Construct the ∆PQR, with PQ = 5.8 cm QR = 4.7 cm and ∠PQR = 60°.

Acme Mathematics 7 299EvaluationTime: 48 minutes Full Marks: 201. A sample square having length 5cm is given below. (a) Prepare the tessellation using the square. [3](b) Write the name of the objects made by the given net. [1]2. (a) From the given pairs of congruent triangles,find the value of x and y. [3](b) Complete the given figure where the dotted line represents the line of symmetry and find out the axes of symmetry. [1]3. A sample rectangle having length 5cm and breadth 3cm is given below.(a) Prepare the tessellation using right angled triangle. [3](b) Write the name of the objects made by the given net. [1]4. Study the figure given along side.(a) Copy the figure and divide it into 4 equal parts. What is the line which divides if called ? [3](b) What is point symmetry ? [1]5. (a) Draw the model figure of cylinder. [3] (b) How many degrees of angle do the line pointing north-east make? [1]mB CAE FD40° 40°60° 60°y5cm4cm 7cmxz

300 Acme Mathematics 76Working hour : 10UNIT Statistics1. Write the number represented by the given tally bars:(a) ||| (b) | (c) |||| (d) ||| (e) ||2. Write the tally bars represented by the following numbers:(a) 17 (b) 20 (c) 13 (d) 18 (e) 293. Study the given marks and fill in the blanks. 19, 10, 15, 11, 22, 25, 10, 11, 22, 19, 15, 25, 22, 11, 10, 11, 19, 10 Marks 11 15 22 25 19 10TallyFrequency .............. .............. .............. .............. .............. ..............4. The ages of students of class 7 of a school is given below. 11 years, 12 years, 13 years, 15 years, 14 years, 13 years, 14 years,12 years, 11 years, 14 years, 15 years, 12 years, 11 years, 11 years,14 years, 15 years, 11 years, 12 years, 11 years, 14 years, Now, answer the following questions: (a) How many students are 12 years old? (b) How many students are more than 10 years? (c) How many students are less than 14 years? (d) How many students are there in the class? 5. Study the given bar graph and answer the questions given below. (a) What is the total number of students from class-4 to class-7 ?(b) Which class has the lowest number of students ? (c) Which class has the highest number of students ? 010203040504ClassesNumber of students5 6 7Warm Up Test