Geometry GEOMETRY 251 iii. Rhombus A rhombus is an equilateral quadrilateral. In other words, a rhombus is a parallelogram with equal adjacent sides. In the adjoining figure, ABCD is a parallelogram with equal adjacent (or consecutive) sides AB and BC. So, ABCD is a rhombus. Its diagonals bisect each other at right angle. iv. Square A square is a regular quadrilateral. In other words, a square is a rectangle with equal adjacent sides. In the adjoining figure, ABCD is a rectangle with equal adjacent sides AB and BC. Hence, ABCD is a square. Here also diagonals AC and BD are equal, but they bisect each other at right angles at O. Properties of Parallelogram Experimental Verification-1 Property 1. The opposite sides of a parallelogram are equal. Step 1: Draw three parallelograms of different shape and size, and label each of them as ABCD. A C B D A D B C A B C D Fig (i) Fig (ii) Fig (iii) Step 2: Measure the pairs of opposite sides AB, DC, AD and BC in cm and fill the results in the table below: Fig AB DC AD BC Remarks (i) (ii) (iii) Conclusion: From the above table, we note that “Opposite sides of a parallelogram are equal.” A B D C C D A B
252 The Leading Mathematics - 7 Experimental Verification-2 Property 2. The opposite angles of a parallelogram are equal. Step 1: A pair of set squares is used to draw a pair of parallel lines l and m. Step 2: A pair of set squares is used to draw another pair of parallel lines p and q intersecting the first pair at A, B and D, C. ABCD is a parallelogram. A D B C Fig (i) A C B D Fig (ii) A B C D Fig (iii) Step 3: The Steps 1 and 2 are repeated to draw two other parallelograms. Each figure is again labeled as ABCD. The figures are denoted as Fig. I, Fig. II and Fig. III respectively. Step 4: In each figure, measure each pair of opposite angles, i.e., ABD and ACD, BDC and CAD. Fill the results in the table below : Fig ∠A ∠C ∠B ∠D Remarks (i) (ii) (iii) Conclusion: From the above table, it is clear that: “The opposite side angles of a parallelogram are equal.” Experimental Verification-3 Property 3. The diagonals of a parallelogram bisect each other. A O B C D A C B D O A C B D O Fig (i) Fig (ii) Fig (iii)
Geometry GEOMETRY 253 Step 1: Three different parallelograms of different size are drawn with the help of a ruler and set-squares and each is labeled ABCD. Step 2: Diagonals AC and BD are drawn so as to intersect at O. Step 3: Measure the lengths of AO and CO and BO and DO in cm. Fill the results in the table below: Fig AO CO BO DO Remarks (i) (ii) (iii) Conclusion: From the above table, it is clear that: “The diagonals of a parallelogram bisect each other.” Properties of Rectangle Experimental Verification-4 Property 4. The diagonals of a rectangle are equal. Step 1: Three different rectangles with different size are drawn with the help of a ruler and set-squares and each is labeled ABCD. Step 2: Diagonals AC and BD are drawn so as to intersect at O. A D B C O Fig (i) A C B D O Fig (ii) A C B D O Fig (iii) Step 3: Measure the lengths of AC and BD in cm. Fill the results in the table below: Fig AC BD Remarks (i) (ii) (iii) Conclusion: From the above table, it is clear that: “The diagonals of a rectangle are equal.”
254 The Leading Mathematics - 7 Properties of Rhombus Experimental Verification-5 Property 5. The diagonals of a rhombus bisect each other at right angles. Step 1: Three different rhombuses of different size are drawn with the help of a ruler and set-squares and each is labeled ABCD. Step 2: Diagonals AC and BD are drawn so as to intersect at O. Fig (i) Fig (ii) Fig (iii) A B C O D A B C O D A B C O D Step 3: Measure the angle AOD, AO, CO, BO and DO. Fill the results in the table below: Fig AO CO BO DO ∠AOD Remarks (i) (ii) (iii) Conclusion: From the above table, it is clear that: “The diagonals of a rhombus bisect each other at right angles.” CLASSWORK EXAMPLES Example 1 Observe the given figure ABCD. (a) Name the figure ABCD. (b) Find the value of x. (c) Find the length of AB and CD. (d) In which condition does the parallelogram ABCD become a rectangle? Solution: (a) the name of the figure ABCD is a parallelogram. (b) We know that the opposite sides of a parallelogram are equal. Since, in the given figure, ABCD is a parallelogram, so D C A 5x – 12 cm B 2x + 3 cm
Geometry GEOMETRY 255 AB = CD or, 5x – 12 = 2x + 3 or, 5x – 2x = 3 + 12 or, 3x = 15 or, x = 15 3 or, x = 5 cm. (c) AB = 5x – 12 = 5 × 5 – 12 = 25 – 12 = 13 cm CD = 2x + 3 = 2 × 5 + 3 = 10 + 3 = 13 cm (d) If any one angle of the parallelogram is a right angle. It becomes a rectangle. Example 2 Observe the adjoining figure. (a) Find the measure of ∠RQT in the given parallelogram PQRS. (b) Are the measures of ∠SPT and ∠ARS equal? Justify by finding their values. Solution: (a) From the given parallelogram PQRS, (i) ∠PQR = ∠PST [ Opposite angles of parallelogram PQRS] or, ∠PQR = 125° (ii) ∠RQT + ∠PQR = 180° [ Angles at the same point of the straight line] or, ∠RQT + 125° = 180° or, ∠RQT = 180° – 125° or, ∠RQT = 55°. R P Q S T 125°
256 The Leading Mathematics - 7 (b) ∠QRS = ∠TQR [Alternate angles] = 55° ∠SPT = ∠TQR [corresponding angles] = 55° Hence, ∠SPT and ∠QRS are equal in measure. EXERCISE 14.2 Your mastery depends on practice. Practice like you play. 1. Write down the names of the quadrilateral in which; (a) one pair of opposite sides are parallel. (b) all sides and all angles are equal. 2. Write the names of quadrilaterals in which; (a) two pairs of opposite sides are parallel. (b) all sides and opposite angles are equal. (c) two pairs of opposite sides and the all angles are equal. 3. (a) Draw two pairs of parallel lines so that each pair is transversal of other pairs. What shape will you get in the middle? (b) Draw one pair of parallel lines and other two transversals that are not parallel and they do not intersect within parallel line. What shape will you get in the middle? 4. In the figure, D, E and F are the midpoints of AB, BC and AC of ∆ABC respectively. (a) Can you name three trapeziums? (b) Can you name three parallelograms? 5. Which of the following statements is true? (a) A rectangle is a parallelogram. (b) All parallelograms are rectangles. (c) A square is also a rhombus. (d) A rhombus is a square. (e) Equiangular and quadrilateral is a rectangle. (f) A square is a rhombus. A D B E F C
Geometry GEOMETRY 257 (g) Equiangular quadrilateral is a square. (h) All squares are parallelograms. 6. Observe the following figures. (a) D C A B x – 3 cm 9 cm (b) D C A B 3x + 1 cm 7x – 15 cm (c) B C A D 3x – 9 cm x + 5 cm (i) Name the given figures. (ii) Find the value of x. (iii) Find the length of AB and CD. 7. Observe the following figures. (a) (b) 2x – 5° x + 25° A D B C (c) S R P Q x + 33° 4x – 57° (i) Write the relation between the opposite angles of a parallelogram. (ii) Find the value of x in each figure. (iii) Find the measures of all the angles of the parallelogram. 8. Study the following figures. (a) (b) (c) O 4x 12 cm y + 5 cm 15 cm A B D C (i) Find the values of x, y and z. (ii) Are the angles x and y equal? S R P Q x 125° C B D A E 123° x y z 72° P Q T S R x y z
258 The Leading Mathematics - 7 9. Observe the following parallelograms. (a) (b) (c) (i) Write the relation between the diagonals of a parallelogram. (ii) Find the values of x, y and z. (iii) Find the measures of the unknown angles of the parallelogram. 10. Observe the given parallelograms. (a) (b) (c) (i) Write the relation between the opposite sides of a parallelogram. (ii) Find the value of 'x'. (iii) Find the length of AC, PR or MK. 11. Observe the following figures. (a) A D B C (b) Q P R S O (c) W Z X Y O AC = 2a BD = 16 cm PR = 3a + 4 cm QS = 7a – 8 cm XO = 4a – 3 cm YO = a + 6 cm (i) Name the figures. (ii) Find the value of 'a'. (iii) Find the actual length of the diagonals. X Y Z W 3x y 6x z 42° 65° x z y A B D C Q R S P 135° y x z 115° 4x – 1 cm x + 2 cm A B D C O 2x – 4 x P S Q R O M L K N O 3x + 13 7x + 1
Geometry GEOMETRY 259 12. Observe the given plane figures. (a) (b) (c) (d) (e) (f) (i) Name the figures. (ii) Find the value of 'x'. (iii) In which condition does a rhombus become a square? ANSWERS 6. (a) 12 cm, 9 cm, 9 cm (b) 4 cm, 13 cm, 13 cm (c) 7 cm, 12 cm, 12 cm 7. (a) 125°; 55°, 125°, 55°, 125° (b) 30°; 55°, 125°, 55°, 125° (c) 30°; 63°, 117°, 63°, 117° 8. (a) 57°, 57°, 57°, yes (b) 72°, 108°, 72°, No (c) 3 cm, 10 cm, No 9. (a) 20°, 120°, 60°; 60°, 120°, 60°, 120° (b) 65°, 42°, 73°, 115°, 65°, 115° (c) 115°, 20°, 45°; 65°, 65° 10. (a) 1 cm, 6 cm (b) 4 cm, 8 cm (c) 3 cm, 44 cm 11. (a) 8 cm, 16 cm (b) 3 cm, 13 cm (c) 3 cm, 9 cm 12. (a) 3 cm (b) 2 cm (c) 0 cm (d) 15° (e) 10° (f) 16° O 4x 12 cm A B D C P S Q R O x + 2 cm 3x–2 cm 3x + 4 cm 4x + 4 cm K L O M N D E C F 30° 2x 3x x + 20° K M L N D C A B 4x + 12° 6x – 20°
260 The Leading Mathematics - 7 14.3 Pythagoras Theorem and Its Use At the end of this topic, the students will be able to: ¾ verify the Pythagoras′ theorem for right-angled triangle. ¾ use the Pythagoras′ theorem in daily life activities. Learning Objectives Note down the following parts of a right-angled triangle. In a right-angled triangle ABC, ∠B = 90°, and ∠C = q is the angle of reference. With reference to the angle q; AB which is opposite to the angle q is perpendicular (p), the side opposite to the right angle is the hypotenuse (h) and the remaining side is the base (b). A B q C Hypotenuse (h) Base (b) Perpendicular (p) Activity 1 Construct a right-angled triangle ABC with sides AB = 5 units, BC = 4 units and AC = 3 units. On AB, BC and AC, construct squares of length equal to AB, BC and AC respectively as in the figure below. On each square, construct unit squares of length 1 unit × 1 unit. Now answer the following questions: a) How many square units does the square with side BC have? b) How many square units does the square with side AC have? c) How many square units does the square with side AB have? d) What is the sum of square units on the square with sides BC and AC? e) Is the sum of squares of BC and AC equal to the square of AB? f) Taking AC = p, BC = b and AB = h. Can you write a true statement containing p, b and h? Activity 2 In each of the following right-angled triangle given below, find the squares of each side and tabulate the results below. A B C 4 5 3
Geometry GEOMETRY 261 A B C 12 cm 13 cm 5 cm Fig. (ii) A B C 13 cm 12 cm 9 cm Fig. (iii) 10 cm 6 cm 8 cm A B C Fig (i) Figure AC2 AB2 BC2 AB2 + BC2 Remarks (i) (ii) (iii) You observed that AC2 = AB2 + BC2.. i.e. h2 = p2 + b2 In any right-angled triangle, the square on the hypotenuse equals to the sum of squares on the other sides. In symbol, h2 = p2 + b2 . This is known as Pythagoras’ theorem. CLASSWORK EXAMPLES Example 1 Observe the given triangle. (a) Write the name of the given figure. (b) If BC = 12 cm and AB = 13 cm, find AC. (c) If ∠B = 40°, what will be the measure of ∠A? Find it. Solution : (a) The given triangle is a right-angled triangle. (b) In right-angled ∆ABC; AB = 13 cm, BC = 12 cm To find AC; we have, by Pythagoras theorem AB2 = AC2 + BC2 or (13)2 = AC2 + (12)2 or AC2 = 169 – 144 or AC2 = 25 b x p h A B 12 cm C 13 cm
262 The Leading Mathematics - 7 or AC2 = (5)2 ∴ AC = 5 cm. (c) If ∠B = 40° and ∠C = 90°, then ∠A + ∠B + ∠C = 180° [ sum of the angles of a triangle] or, ∠A + 40° + 90° = 180° or, ∠A + 130° = 180° or, ∠A = 180° – 130° or, ∠A= 50° Example 2 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, find the length of the ladder. Solution : Since the wall stands vertically with the ground, the problem can be solved by using right-angled triangle of sides 6 ft and 8 ft. In right-angled ∆ABC, AB = 8 ft, BC = 6 ft and AC = ? By Pythagoras' theorem, AC2 = AB2 + BC2 or, AC2 = 82 + 62 or, AC2 = 64 + 36 = 100 or, AC2 = 102 ∴ AC = 10 ft Hence, the length of the ladder is 10 ft. EXERCISE 16.3 Your mastery depends on practice. Practice like you play. 1. Find the length of the missing side in each of the following right-angled triangles. (a) A B C 40° 9 cm 12 cm (b) A B 6.4 cm C 4.8 cm 45° (c) B 8 cm C A 50° 6 cm 8 ft 6ft wall A B C 8ft 6ft ?
Geometry GEOMETRY 263 (d) A C B 35° 17 cm 8 cm (e) A 60° B C 41 cm 9 cm (f) A C B 55° 25 cm 24 cm (g) A B C 70° 3 cm 3.4 cm (h) A B 30° C 7 cm 15 cm (i) C A B 48° 5.2 cm 2 cm 2. Given below are the length of sides of a triangle. Determine whether or not the sides make a right-angled triangle. (a) 3 cm, 4 cm, 5 cm (b) 4 cm, 5 cm, 6 cm (c) 5 cm, 12 cm, 13 cm (d) 6 cm, 8 cm, 10 cm (e) 4 cm, 4 cm, 4 2 cm (f) 5 cm, 6 cm, 10 cm (g) 10 cm, 24 cm, 26 cm (h) 9 cm, 12 cm, 15 cm (i) 7 cm, 8 cm, 9 cm 3. (a) A ladder rests against a wall of height 12 ft. The foot of the ladder is 5 ft away from the foot of the wall. (i) How much degrees of angle does the wall make with the ground? (ii) Find the length of the ladder. (iii) If the ladder makes an angle 60° with the ground, which angle does it make with the wall? Find it. (b) A wire of the length 10 m supports an electricity pole. If the one end of the wire is fixed at a distance of 6 m away from the foot of the pole, (i) Write the measure of angle made by the electric pole to the ground. (ii) Find the height of the pole. (iii) If the wire makes 50° with the pole, find the measure of the angle made by the wire with the ground level. 12 ft 5ft A B C 10 m 6 m
264 The Leading Mathematics - 7 4. (a) Figure shows the gate of length 8 m and height 6 m. (i) Which angle does the vertical frame make with the horizontal frame? (ii) How long is the diagonal of the gate? (b) A 10 ft long ladder rests against a wall so that its foot is 6 ft away from the foot of the wall. (i) How high is the wall? (ii) If the ladder makes an angle 60° with the wall, how much degrees of angle does it make with the ground level? Find it. 5. (a) A 13 ft long ladder rests against a wall of height 12 ft. (i) How far is the end of the ladder from the foot of the wall? (ii) What is the sloping angle of the ladder if it makes 30° with the wall? (b) Figure shows the children’s slide. If the sliding plate is 26 ft long and the lower end of it is 24 ft away from the foot of the supporting pillar, how high is the pillar? 6. (a) Estimate the length of the triplate needed to roof the house using the information given in the figure. 8 ft ? 6 ft 6 ft 10 ft 12 ft 13 ft 26 ft 24 ft 1ft 8ft 6ft 1ft
Geometry GEOMETRY 265 (b) Find the length of the slope of conical lid of the height 16 cm of the cylindrical tank given in the figure alongside. 7. (a) Figure is a cube of side 6 cm. (i) Find the length of the diagonal of the base square. (ii) Work out the length of its inner diagonal. (b) Figure shows the model of a match box. Work out the length of face diagonals and inner diagonal. 8. (a) Find the height of the isosceles triangle having length of equal sides = 13 cm and base = 10 cm. (b) Find the height of an equilateral triangle of side 6 cm. ANSWERS 1. (a) 15 cm (b) 8 cm (c) 10 cm (d) 15 cm (e) 40 cm (f) 7 cm (g) 1.6 cm (h) 13.2665 cm (i) 4.8 cm 2. (a), (c), (d), (e), (g) and (h) form right-angled triangles. 3. (a) 90°, 13 ft, 30° (b) 90°, 8 m, 40° 4. (a) 90°, 10 ft (b) 8 ft 5. (a) 5 ft, 60° (b) 10 ft 6. (a) 12 ft (b) 22 cm 7. (a) 8.49 cm, 10.39 cm (b) 6.71 cm, 5 cm, 7.21 cm, 7.81 cm 8. (a) 12 cm (b) 5.20 cm 24 cm 1 cm 1 cm Q. No. 7 (a) 6 cm 6 cm 6 cm 4 cm 3 cm Q. No. 7 (b) Q. No. 8 (a) A B D C 13 cm 13 cm A B C D 6 cm Q. No. 8 (b)
266 The Leading Mathematics - 7 CHAPTER 17 Congruency Lesson Topics Pages 17.1 Congruency 267 What are the shape and size of your both hands? Alike or different? What are the shape and size of your notebooks? What are the shape and size of your pencil and erasers and your friends? Whose have the same and whose have different? Are the squares with the same and different sides equal? WARM-UP
Geometry GEOMETRY 267 17.1 Congruency At the end of this topic, the student will be able to: ¾ define similar and congruent geometric figure. Learning Objectives Congruent Triangles In everyday life, we talk of twins. They are supposed to look alike both in shape and size. Today, modern science has gone beyond that. It has developed a technique known as cloning. By this technique, two living beings who look alike in all respects can now be produced or developed. In the study of geometry, we begin by accepting two points always alike or identical or one point can be fitted exactly onto another. In other words, we say that two points are always congruent. Then, coming to the case of a line segment, we say two line segments are said to be congruent if they have the same length. As regards angles, two angles are said to be congruent, if each of them has the same measure. In high school geometry, we are mostly concerned with plane figures such as triangles, quadrilaterals, circles, etc. It is easy to visualize how a circle can be fitted or placed exactly over another if both have the same radius. Similarly, it is not difficult to see how a square can be placed exactly over another square having equal side. But, in the case of triangles, fitting one triangle with another or superposing of triangles can be done by drawing triangles on paper, cutting them out and fitting them together. Such activities or experiences of daily life lead us to what is known as congruence of geometrical figures. Now onwards, we shall agree with the following definition: If two geometrical figures fit exactly on each other, they are said to be congruent or in congruence. Testing the congruency of two figures or superposing of one figure onto another will be done only in the following three ways: 1. sliding or translating in the plane without rotation, 2. rotating about a point in the plane, or 3. inverting upside down (without stretching and tearing). Fig. (i) ≅ Fig. (ii) Fig. (iii) ≅ Fig. (iv)
268 The Leading Mathematics - 7 In each case, it is better to start with a fixed point such as endpoint of a line segment, vertex of an angle, centre of a circle. Conditions for Congruent Figures Name Figures Conditions Notation Line Segments A B C D 3.4cm 3.4cm AB = CD = 3.4 cm AB ≅ CD Angles 50° A O B 50° P O Q ∠AOB = ∠POQ = 50° ∠AOB ≅ ∠POQ Triangles A B C P Q R AB = PQ, BC = QR, AC = PR, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R ∆ABC ≅ ∆PQR Square K N L M A D B C LM = BC KLMN ≅ ABCD Rectangles S R P Q A D B C AB = SP, BC = PQ ABCD ≅ PQRS Circles A B P Q Radius AB = Radius PQ Circle A ≅ Circle P EXERCISE 17.1 Your mastery depends on practice. Practice like you play. 1. Observe the following pairs of the geometric shapes. (a) A (b) B C D E F A B C D E F
Geometry GEOMETRY 269 (c) (d) (e) (f) (g) (h) (i) Measure their necessary parts for using conditions. (ii) Which of the following pairs of figures are congruent by using conditions? 2. Study the following pairs of the shapes. Which of the following pairs of figures are congruent ? Why ? (a) P Q 5 cm 3 cm 6 cm C A 5 cm 3 cm 6 cm B C (b) P Q C A 75° 60° 75° 45° 60° 45° B C (c) (d) ANSWERS Consult with your teacher. A B C D A O B P Q R K N L M A D B C S R P Q A D B C A B D C A B D C A B P Q 5 cm 5 cm 5 cm 5 cm 5 cm 5 cm 5 cm 5 cm K N L M A D B C A E A E B B C C D D 6 cm 6 cm 7 cm 7 cm 5 cm 4 cm 5 cm 4 cm 4 cm 4 cm
270 The Leading Mathematics - 7 CHAPTER 18 Solid Objects Lesson Topics Pages 18.1 Skeleton Model of Polyhedron 271 18.2 Relation Among Parts of Polyhedron 274 What is solid ? What is the meaning of 3-dimensianl figure? What is the shape of your duster ? What is the shape of the box ? What is the shape of the ball ? Why is the cube different from cuboid ? What are faces, edges and vertices of the solid ? How many faces, edges and vertices are in Rubik ? How many faces, edges and vertices are in duster ? Draw the nets of cube, cuboid and triangular prism. WARM-UP
Geometry GEOMETRY 271 18.1 Skeleton Model of Polyhedron At the end of this topic, the student will be able to: ¾ prepare the skeleton models of tetrahedron, octahedron, cone and cylinder. Learning Objectives Introduction We use models to visualize geometric concepts such as peak of needle to visualize point, straight thread to visualize line and sheet of paper to visualize the concept of a plane. A plane is two dimensional flat surface with no thickness which extends infinitely in all direction. Similarly, a cuboid model is used to visualize space. Space is a three dimensional solid which extends infinitely in all directions. Examples of solid shapes include cubes, pyramids and spheres. Solid shapes are classified by their regularity according to number of faces, edges and vertices they have, and the shape of their faces. Faces are flat sides on a solid shape. Edges are the straight locations where two faces come together. A vertex is a pointy place where more than two faces meet. Some solid shapes only have curves and irregularities instead of faces, edges and vertices. One example is the sphere. Some shapes have a combination of faces and curved parts, such as cylinders and cones. cube sphere rectangular prism pyramid cylinder cone Skeleton Model of Solid Shapes Skeletons mean the structural frame of bodies. For examples; the bone system of human body is most familiar skeleton model. Similarly, the trust of roof, towers of bridge, structural map of building, hydro-power, etc are the skeleton models. In the mathematics, the frame that shows the edges and vertices of a solid shape is called a skeleton of the solid. Skeleton shapes of the geometrical bodies (solid) are made with balls of modeling clay and straws, juice pipe, Nigalo or wheat straw (Chhwali), etc. Specially, we use the sticks joining by the rope as shown in the above figure. The adjoining figure shows a cube and a skeleton cube:
272 The Leading Mathematics - 7 How many balls of modeling clay and how many straws does it take to make the cube? The skeleton models of some geometric solid in the table given below: Name of Solid Geometric Shape Skeleton Model Cube Cuboid Tetrahedron Pyramid Octahedron Cone
Geometry GEOMETRY 273 Name of Solid Geometric Shape Skeleton Model Cylinder EXERCISE 16.1 Your mastery depends on practice. Practice like you play. 1. Draw the geometric figures of the following solid shapes: cube rectangular prism pyramid cylinder cone 2. Draw the skeleton model of the following geometric shapes: Cone Square-based pyramid Tetrahedron (triangle-based pyramid) Triangular prism Cube Cuboid Cylinder Hexagonal prism ANSWERS Consult with your teacher.
274 The Leading Mathematics - 7 16.2 Relation Among Parts of Polyhedron At the end of this topic, the students will be able to: ¾ identify the physical solid shapes and their faces, edges and corners. Learning Objectives A geometrical figure whose points cannot lie in a single plane is called a solid shape or solid figure. For example, if the four points A, B, C and D lie on the same plane α, they form a quadrilateral. But, the point D does not lie in the same plane α, it lies in another plane β. What happen? They form a triangular pyramid or tetrahedron as shown in the adjoining figure. This type of slid is called a polyhedron (Plural; polyhedra). Look at the familiar solids and their geometrical shape and name below: Physical Shape Geometrical Shape Name of Solid Cuboid Cube Cylinder Cone Sphere Pyramid A A B B C C D D α α β
Geometry GEOMETRY 275 Face, Edge and Corner (Vertex) of Solid Shape Look at the following solids and discuss about their faces, edges and corners or vertices: Now, define the faces, edges and corners of the solid shape below: Face: The triangular or rectangular surface or planes of a solid shape (polyhedron) is called its face. Edge: The line segment formed by intersecting the faces or planes of a solid shape is called its edge. Corner: The point formed by intersecting the edges of a solid shape is called its corner or vertex. Look at the following solids and discuss about the number of faces, edges and vertices: Cube Cuboid Prism Tetrahedron Pyramid Identify the number of faces, edges and vertices of the above mentioned solids and fill the table given below: Polyhedron No. of Faces (F) No. of Vertices (V) No. of Edges (E) F + V - E Result Cube 6 8 12 6 + 8 – 12 = 2 F + V – E = 2 Cuboid 6 8 12 ......... ......... Prism 5 6 9 ......... .........
276 The Leading Mathematics - 7 Polyhedron No. of Faces (F) No. of Vertices (V) No. of Edges (E) F + V - E Result Tetrahedron 4 4 6 ......... ......... Pyramid 5 5 8 ......... ......... From the above table, we conclude that the relation F + V – E = 2 is satisfied. This relation is known as Euler's formula. In other hand, count the faces, edges and vertices of cylinder, cone and sphere, and test the Euler's formula. What do you find? Is the relation F + V – E = 2 satisfied? Polyhedron No. of Faces (F) No. of Vertices (V) No. of Edges (E) F + V - E Result Cylinder 3 0 2 3 + 0 – 2 = 1 F + V – E = 1 Cone 2 1 1 ......... ......... Sphere 1 0 0 ......... ......... Hence, the Euler's relation is only satisfied in the solid shapes having the plane surfaces. EXERCISE 16.2 Your mastery depends on practice. Practice like you play. 1. Write the geometric name of the following solid shapes: (a) (b) (c) (d) (e) (f) Cylinder Cone Sphere
Geometry GEOMETRY 277 (g) (h) (i) 2. Draw the geometrical shape of the solids mentioned in the Q.No.1. 3. Observe the following solids. (i) Write down the number of faces, edges and vertices of the given geometric shapes. (ii) Test the Euler's formula in the table. (iii) Is the relation Euler's formula satisfied for the all shapes? (a) (b) (c) (d) (e) (f) (g) (h) ANSWERS Consult with your teacher.
278 The Leading Mathematics - 7 CHAPTER 17 Coordinates Lesson Topics Pages 17.1 Introduction to Coordinates 279 17.2 Horizontal and Vertical Distances 285 What is number line ? Can you draw a number line ? Read and write numbers of number line. What are perpendicular lines ? Can you draw perpendicular lines ? Where are the negative and positive numbers on the graph paper ? Can you count the square grids on the graph paper ? What are the coordinates of the points A, B, C and D in the above graph? What are the positions of the students in the class? Discuss the uses of coordinates in our daily life activities. WARM-UP O X Y A B C D
Geometry GEOMETRY 279 17.1 Introduction to Coordinates At the end of this topic, the students will be able to: ¾ write the coordinates of the points from the graph. ¾ plot the points in the graph. Learning Objectives In the previous grade, we learned about how to locate the position of a point on a line and on a plane. Numbers are used to represent geometric points on a line. Ordered pair of numbers is used to represent a point on a plane. Geometric objects can be defined and interpreted in terms of numbers and number pairs called coordinates. This approach was initiated for the first time by the French mathematician René Descartes (1596-1650). Coordinates of Points in Graph We have already seen how a number line can be used to denote a single point on a line and a point on a line can be used to represent a number. Now, we see how an ordered pair of numbers can be used to denote a point in a graph and conversely. First, we draw a horizontal number line (a line drawn parallel to the base margin of the book). We denote it by XOX' and name it as the x-axis. Then, we draw another vertical number line YOY' (perpendicular to the horizontal line XOX' and meeting XOX' at the point O). The line YOY' is called the y-axis. The common point O is called the origin. The positive directions are indicated by the arrow-heads. In other words, the directions towards the right and upwards are accepted as positive directions; and those towards the left and downwards are taken to be negative. In each of these axes, a suitable scale (or unit of length) is fixed. Then, starting from the origin, points at equal distance on each axis are marked. As usual, we associate i) the number zero to the origin, ii) the numbers 1, 2, 3, … to successive points towards the right and above, and iii) the numbers –1, –2, –3, … to the successive points towards the left and below the origin. O M x K P(x, y) X′ X (0, 0) y-axis x-axis Y Y′ y
280 The Leading Mathematics - 7 The decimal numbers are located in the same way as in the case of the number line. Now, we can locate any point on the plane by means of an ordered pair of numbers and conversely. Consider a point P in the above plane. Draw perpendiculars PM and PK from P to the coordinate axes. Let them meet the x-axis at M and the y-axis at K. Denote the length of the or directed line segment OM, |OM| by x. This is called the x-coordinate or abscissa of the point P. In the same way, denote the length of the directed line segment OK, |OK| by y. This is called the y-coordinate or ordinate of the point P. Remember that both lengths are usually expressed in the same unit. In practice, the two numbers are written as an ordered pair (x, y) with the x-coordinate written first in the parenthesis ( ). It is easy to see that the ordered pair (x, y), the coordinates of P, are sufficient to locate the position of P in the plane. To do so, we start at the origin, move along the x-axis a distance of x-units (towards the right if x is positive and towards the left if it is negative), and then parallel to the y-axis through y-units (upwards if y is positive and downwards if it is negative). We then arrive at the desired point P. The above system of assigning an ordered pair of numbers to a point in a plane and conversely representing a point in a plane by an ordered pair of numbers (i.e., the above system of coordinating an ordered pair to every point in the plane) is called the Rectangular or Cartesian Coordinate System. The plane so obtained is called the Cartesian plane after Rene Descarte, the father of coordinate geometry. The Cartesian coordinate system divides the whole plane into four similar sections called quadrants. The four quadrants are characterized by the following signs-convention: In short, we have Quadrant I x > 0, y > 0 or (+, +) Quadrant II x < 0, y > 0 or (–, +) Quadrant III x < 0, y < 0 or (–, –) Quadrant IV x > 0, y < 0 or (+, –) O X+ X Y Y+ x > 0, y > 0 Quadrant I (+, +) x > 0, y < 0 Quadrant IV (+, –) x < 0, y > 0 Quadrant II (–, +) x < 0, y < 0 Quadrant III (–, –)
Geometry GEOMETRY 281 CLASSWORK EXAMPLES Example 1 Given the ordered pairs of numbers A(3, 4) and B(4, –3). Represent them by points in a grid or a squared paper or graph. Solution: We consider XOX’ and YOY’ as two axes in the squared paper. Fix O as the origin. Then, (i) To plot or locate the point A(3, 4). We start at O on the graph, Take 1 unit = 1 big square or 5 small squares Move 3 units rightwards to L; Move up 4 units up to a point A say. A is the point representing the ordered pair (3, 4). (ii) To plot or locate the point B(4, –3); We start at O, Take 1 unit = 1 big square or 5 small squares Move 4 units rightwards to M; Move up 3 units down to a point B say. Then, B is the point (4, –3). Example 2 Observe the two points P and Q on the graph. (a) Draw vertical lines PL and QM meeting OX and OX' at L and M respectively. (b) Write the coordinates or P and Q. Solution : (a) Draw PL and QM meeting OX and OX' at L and M respectively. Count the number of units in OL and LP. O 3 4 4 3 L M B(4, –3) A(3, 4) X Y O 2 4 L Q(–2, 4) P(2, 6) X' X Y' Y 6 M –2
282 The Leading Mathematics - 7 Here, OL = 2 units and LP = 6 units. Count the number of units in OM and MQ. Here, OM = – 2 units and MQ = 4 units (b) The coordinates of P and Q are (2, 6) and (–2, 4) respectively. Example 3 Observe the given points P(3, 6), Q(3, 3) and R(7, 3). (a) Plot the points P(3, 6), Q(3, 3) and R(7, 3) on the graph. (b) Join PQ, QR and RP. (c) Name the plane figure that you get. Solution: (a) Locate the origin O and then move 3 units right on x axis. And then move 6 units up on y-axis for the point P(3, 6). Similarly, plot the other points Q(3, 3) and R(7, 3). O X Y P Q R (b) Then join PQ, QR and RP. (c) The name of the plane figure is a triangle. EXERCISE 17.1 Your mastery depends on practice. Practice like you play. 1. Plot the following points on the squared grid paper or graph. (a) (1, 0) (b) (0, 1) (c) (−1, 0) (d) (0,−1) (e) (1, 1) (f) (2, 2) (g) (3, 3) (h) (4, 4) (i) (1, −1) (j) (−2, 2) (k) (3, 4) (l) (4, 3) (m) (5, 4.5) (n) (2.5, 4) (o) (0, −10) 2. Plot the points in the same grid or squared paper: (a) (−1, −1), (2, 2), (3, 3) and(4, 4) (b) (2, 4), (– 2, 3), (–3, – 1) and (2, 0)
Geometry GEOMETRY 283 3. Observe the given points. (a) P(−3, −4), Q(3, −4) and R(3, 4). (b) P(−5, 0), Q(5, 0) and R(0, 5), (c) P(0, 0), Q (3, −4) and R(– 3, − 4). (i) Plot the points P, Q and R in the same grid or squared paper. (ii) Join PQ, QR and RP in each case. (iii) Write the name of the plane figure that you get. 4. Study the given points and answer the questions. A(1, 2), B(4, 2), C(3, 6) and D(6, 6) (a) Plot the points A(1, 2), B(4, 2), C(3, 6) and D(6, 6). (b) Join AB, BC, CD and DA on the graph. (c) Write the name of the plane figure so formed. 5. Observe the given points and answer the questions. K(0, 0), L(5, 0), M(5, 5) and N (0, 5) (a) Plot the points K(0, 0), L(5, 0), M(5, 5) and N (0, 5). (b) Join KL, LM, MN and NK on the graph. (c) Write the name of the plane figure so formed. 6. Observe the following graphs. (a) (b) (i) Read the coordinates of the corners or vertices of the figures. (ii) Write their coordinates. O X Y A B C D O X Y A B C D
284 The Leading Mathematics - 7 7. Observe the following graphs. (a) (b) (i) Read the coordinates of the corners or vertices of the figures. (ii) Write their coordinates. 8. Study the following graphs and answer the questions. (e) (f) (i) Read the coordinates of the corners or vertices of the figures (ii) Write their coordinates. ANSWERS Consult with your teacher. O X Y P Q R S O X Y P Q R S O X Y P Q S T R O X Y B C E D F A
Geometry GEOMETRY 285 17.2 Horizontal and Vertical Distances At the end of this topic, the student will be able to: ¾ find the vertical and horizontal distances between two points, ¾ find the mid-point of the horizontal and vertical line segments, ¾ find the coordinates of missing point of the rectangle and square. Learning Objectives Horizontal Distance between Two Points A line parallel to the x-axis is called vertical line. In the adjoining graph, what are the coordinates of A, B, C and D? What is the distance between A and B? How? The distance between the points A and B is the length of the line segment AB, which is equal to 4 units. Similarly, what is the length of the line segment CD? Let's suppose the horizontal line segment joining points P(x1, y1) and Q(x2, y1) in the coordinate plane since for horizontal line, the y-coordinates are the same, where x2 > x1. So, the distance between these two points is given by, d = x2 – x1. i.e., PQ = x2 – x1 if x2 > x1 and PQ = x2 – x1 if x1 > x2. Vertical Distance between Two Points A line parallel to the y-axis or perpendicular to the horizontal line is called horizontal line. In the adjoining graph, what are the coordinates of A, B, C and D? What is the distance between A and B? How? O X Y P(x1, y1) Q(x2, y1) x2 – x1 X Y Y' X' A B C D (1, 2) (2, –1) (5, 2) (5, –2) I know, 5 – 1 = 4 for both AB and CD. But, in horizontal distance, the y-coordinates are the same. X Y Y' X' A B C D (1, 4) (5, 3) (1, 1) (5, –1) O'
286 The Leading Mathematics - 7 The distance between the points A and B is the length of the line segment AB, which is equal to 3 units. Similarly, what is the length of the line segment CD? Let's suppose the horizontal line segment joining points P(x1, y1) and Q(x1, y2) in the coordinate plane since for horizontal line, the y-coordinates are same, where y2 > y1. So, the distance between these two points is given by, d = y2 – y1. i.e., PQ = y2 – y1 if y2 > y1 and PQ = y1 – y2 if y1 > y2. Mid-point of Horizontal and Vertical Line Segments A point which divides the line segment into two equal parts is called a mid-point. In the adjoining graph, the length of the horizontal line segment AB joining the points A(1, 2) and B(5, 2) is 4 units. How can we divide the line segment AB into two equal parts? 2 is the half of 4 and 4 = 5 – 1. So, 2 = ½ (5 – 1), and 2 + 1 = 3 is the x-coordinate. Also, 3 is the average of the 1 and 5, i.e., 3 = ½ (1 + 5) and the y-coordinate is 2 being constant. The mid-point P lies at P as shown in the above figure. Therefore, the coordinates of the mid-point P of AB is (3, 2). Similarly, which would the mid-point of CD be in the same graph? Let us suppose the horizontal line segment joining points A(x1, y1) and B(x2, y1) in the coordinate plane. So, the x-coordinate of the mid-point P of AB is (x, y), where x = ½(x1 + x2) and y = y1. i.e., P(x, y) = P x1+ y2 2 , y1 . For the vertical line segment joining points C(x1, y1) and D(x1, y2), the x-coordinate of the mid-point Q of CD is (x, y), where x = x1 and y = ½ (y1 + y2). i.e., Q(x, y) = Q x1, x1+ y2 2 . O X Y P(x1, y1) Q(x1, y2) y2 – y1 I know, 4 – 1 = 3 for both AB and CD. But, in vertical distance, the x-coordinates are the same. X Y Y' X' A 1 2 2 2 B C D O (1, 2) (3, 4) (5, 2) (3, –2) O X Y A(x1, y1) B(x2, y1) O X Y C(x1, y1) D(x1, y2) Q
Geometry GEOMETRY 287 CLASSWORK EXAMPLES Example 1 Find the distance between the points A(3, 4) and B(–1, 4) and the coordinates of the mid-point of AB. Solution: Here, the given two points are A(3, 4) and B(-1, 4). Now, we know that The required distance AB (d) = x2 – x1 = 3 – (– 1) = 3 + 1 = 4 units. Again, the mid-point of AB, (x, y) = x1+ y2 2 , y1 = –1 + 3 2 , 4 = 2 2 , 4 = (1, 4). Example 2 If the three vertices of a rectangle are (3, –2), (3, 3) and (–1, 3), find its fourth vertex by using graph. Solution: Here, the three vertices of rectangle are (3, 1), (3, 3) and (–1, 3) and plot these vertices on the graph and complete the rectangle by equating the opposite sides. So, the required coordinate of the fourth vertex of the rectangle is (–1, –2). EXERCISE 17.2 Your mastery depends on practice. Practice like you play. 1. Find the horizontal distance between the following pairs of points: (a) A(1, 3) and B(5, 3) (b) C(–2, 4) and D(3, 4) (c) E(4, –2) and F(1, –2) (d) G(–3, 1) and H(–2, 1) 2. Find the vertical distance between the following pairs of points: (a) P(1, 1) and Q(1, 4) (b) R(–2, –1) and S(–2, 4) (c) T(4, –2) and U(4, –5) (d) V(3, 1) and W(3, –5) 3. Find the length of the line segment joining the following pairs of points: (a) L(2, 1) and M(4, 1) (b) I(–12, –4) and J(–12, 14) (c) K(10, –2) and N(10, –4) (d) G(–7, –9) and H(7, –9) 4. Find the midpoint of the line segment joining the following pairs of points by using graph: (a) A(2, 1) and B(4, 1) (b) P(–2, 4) and Q(2, 4) (c) C(1, –3) and D(1, 5) (d) R(3, –1) and S(3, –5) X Y Y' X' O (3, –2) (–1, 3) (3, 3)
288 The Leading Mathematics - 7 5. Which point bisects the line segment joining the following pairs of points? (a) P(12, 10) and Q(14, 10) (b) R(–24, 40) and S(30, 40) (c) D(10, –30) and C(10, 50) (d) B(30, –10) and A(30, –50) 6. (a) If (1, 2), (1, 6) and (7, 6) are three vertices of a rectangle then find the fourth vertex of the rectangle by using graph paper. (b) If (1, –3), (3, –3) and (3, 1) are three vertices of a rectangle then find the fourth vertex of the rectangle by using graph paper. (c) If (0, 2), (–2, 0) and (1, –3) are three vertices of a rectangle then find the fourth vertex of the rectangle by using graph paper. (d) Three vertices of a square are (2, 6), (6, 6) and (6, 2). Plot these vertices in the square paper and find the fourth vertex. 7. (a) Observe the points (1, 3), (4, 0), (6, –1) and (1, –3). (i) Plot the points on the same graph. (ii) What type of figure do you find? Name it. (iii) Find the mid-points of its diagonals. (iv) Are the midpoints of the diagonals the same? Why? (b) Observe the points P(0, 0), Q(4, 0), R(4, 4) and S(0, 4). (i) Plot the points on the same graphs. (ii) What type of plane figure do you see? Name it. (iii) Find the mid-points of its diagonals PR and QR. (iv) Are the midpoints of the diagonals PR and QR the same? Give reason. ANSWERS 1. (a) 4 (b) 5 (c) 3 (d) 1 2. (a) 3 (b) 5 (c) 3 (d) 6 3. (a) 2 (b) 18 (c) 2 (d) 14 4. (a) (3, 1) (b) (0, 4) (c) (1, 1) (d) (3, – 3) 5. (a) (13, 10) (b) (3, 40) (c) (10, 10) (d) (30, – 30) 6. (a) (7, 2) (b) (1, 1) (c) (3, – 1) (d) (2, 2) 7. (a) Rectangle, (–2, 0), Yes (b) Square, (2, 2), Yes Project Work 17.2 Roll two die with numbered 0 to 5 at successively four times, one is red which represents the number along x-axis and another is black which represents the number along y-axis. List the outcomes and plot these points on the graph. Join these points on the same graph and colour it. Name the figure do you get.
Geometry GEOMETRY 289 CHAPTER 18 Symmetry and Tessellation Lesson Topics Pages 18.1 Symmetry 290 18.2 Tessellation 293 What are vertical, horizontal and slanting lines ? Draw a rectangle, square, oval and circle. Fold a rectangular paper sheet once vertically then unfold it. What do you see ? Fold a rectangular paper sheet twice vertically and horizontally then unfold it. What do you see ? Fold a square paper sheet once diagonally then unfold it. What do you see ? Fold a rectangular paper sheet twice diagonally then unfold it. What do you see ? Fold a circular paper four times along diameter and radius, then unfold it. What do you see ? WARM-UP
290 The Leading Mathematics - 7 18.1 Symmetry At the end of this topic, the students will be able to: ¾ introduce line symmetry and point symmetry. Learning Objectives A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn’t change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflection symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other. The simplest case of reflection symmetry is known as bilateral symmetry. For example, each of the following figures exhibits bilateral symmetry: Fig : (i)' Fig : (ii)' Fig : (iii)' Fig : (iv)' The heart and smiley each have a vertical axis of symmetry, and the lobster has a horizontal axis of symmetry. The arrow has an axis of symmetry at an angle. If you draw the reflection line though any one of these figures, you will notice that for every point on one side of the line there is a corresponding point on the other side of the line. If you connect any two corresponding points with a segment, that segment will be perpendicular to the axis of symmetry and bisected by it (cut into two equal length segments): Fig : (i)' Fig : (ii)' Fig : (iii)' Fig : (iv)'
Geometry GEOMETRY 291 An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. If points on a figure are equally positioned about a central point, then we say the object hasrotational symmetry. A figure with rotational symmetry appears the same after rotating by some amount around the center point. The angle of rotation of a symmetric figure is the smallest angle of rotation that preserves the figure. For example, the figure on the left can be turned by 180° (the same way you would turn an hourglass) and will look the same. The center (recycle) figure can be turned by 120°, and the star can be turned by 72°. For the star, where did 72° come from? The star has five points. To rotate it until it looks the same, we need to make 1 5 of a complete 360° turn. Since 1 5 × 360° = 72°, this is a 72° angle rotation. 180° 120° 120° 120° 72° 180° Fig : (i)' Fig : (ii)' Fig : (iii)' Fig : (iv)' Using degrees to describe the rotation amount is inconvenient because the precise angle is not obvious from looking at the figure. Instead, we will almost always use the order of rotation to describe rotational symmetry. Among them, a rotational symmetry turned by 180o at a point is called point symmetry. The fig (v)' and fig (viii)'are point symmetries. Fig : (v) Fig : (vi) Fig : (vii) Fig : (viii)
292 The Leading Mathematics - 7 EXERCISE 18.1 Your mastery depends on practice. Practice like you play. 1. Draw the symmetric line in the following figures: (a) (b) (c) (d) (e) (f) (g) (h) 2. Complete the following : 3. Find the order of the following rotational symmetry. Which of them are point symmetries? (a) (b) (c) (d) (e) (f) (g) (h) ANSWERS Consult with your teacher.
Geometry GEOMETRY 293 18.2 Tessellation At the end of this topic, the students will be able to: ¾ introduce and decorate the tessellation made by triangles. Learning Objectives Atessellation of a flat surface is the tiling of a plane by using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some common tessellations are given below which are used on wallpapers or carpets: Types of Tessellation There are three types of tessellation. Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons. Name Polygon Pattern Tessellation or Tile a tessellation of triangles a tessellation of squares a tessellation of hexagons Regular Tessellation A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides and angles of the polygon are all equivalent
294 The Leading Mathematics - 7 (i.e., the polygon is both equiangular and equilateral). Congruent means that the polygons that you put together are all the same size and shape. Some regular triangular tessellations are given below: Regular Tessellation of Regular Hexagons Regular Tessellation of Equilateral triangles Triangular Tessellation in 3D shaped EXERCISE 18.2 Your mastery depends on practice. Practice like you play. 1. Identify the triangular tessellations.: (a) (b) (c) (d) (e) (f) Symmetry and Tessellation Geometry 229 you put together are all the same size and shape. Some regular triangular tessellations are given below: Practice - 5.6 (B) 1. Identify the triangular tessellations.: (a) (b) (c) (d) (e) (f) Regular Tessellation of Regular Hexagons Regular Tessellation of Equilateral triangles Triangular Tessellation in 3D shaped Symmetry and Tessellation Geometry 229 you put together are all the same size and shape. Some regular triangular tessellations are given below: Practice - 5.6 (B) 1. Identify the triangular tessellations.: (a) (b) (c) (d) (e) (f) Regular Tessellation of Regular Hexagons Regular Tessellation of Equilateral triangles Triangular Tessellation in 3D shaped
Geometry GEOMETRY 295 2. Trace out the following tessellations on your note book and expand twice with colour: (a) (b) (c) 2. Complete the following tessellations with suitable different colours: (a) (b) (c) (d) 3. Make tessellations from the following and colour them: (a) Equilateral Triangle (b) Isosceles Triangle (c) Regular Triangle (d) Mixed Triangles ANSWERS Consult with your teacher.
296 The Leading Mathematics - 7 CHAPTER 19 Transformation Lesson Topics Pages 19.1 Introduction 297 19.2 Reflection 301 19.3 Translation 307 What is the meaning of transformation? Define the transformation. Tell the types of transformation. Tell some examples of reflection, translation and rotation. X X' l Y' Z Y B B WARM-UP
Geometry GEOMETRY 297 19.1 Introduction At the end of this topic, the students will be able to: ¾ introduce the transformation and its types. Learning Objectives An apple falling from a tree involves a change of place from the tree to the ground or a movement. The image of the Machhapuchhre Himal in Phewa Tal looks upside down; but a man looking at a mirror finds his left hand on the right side. The hour hand of a clock rotates about the centre from the 12 hours mark to the 3 hour mark in three hours. Image of a girl Hour clock Reflection of Machhapuchhre in Phewa A girl standing in the Sun some hours before the noon finds her image larger than herself. These are some of the familiar natural phenomena. All these involve some kinds of change or transformation. Ordinarily, to transform is to change or to convert. In the first example, the shape and size of the apple is not changed. It does not turn or rotate. In the second type, the side changes from left to right to left or from top to bottom to top. There is no change in shape or size. Thirdly, there is a rotation about a fixed point. Here also, there is no change in shape and size. The last one is different from the first three. Here, the shape remains the same but the size is changed, i.e., becomes large or small. falling apple
298 The Leading Mathematics - 7 These four types of changes or transformations are given the names: (a) Translation (b) Reflection (c) Rotation, and (d) Enlargement or Reduction. We give below some more diagrams to make the above terms more familiar. (a) Translation: after translation R R before translation (b) Reflection: SATURDAY FEBRUARY 15 SATURDAY FEBRUARY 15 Mirror R R before reflection mirror line after reflection (c) Rotation: R R angle = 90° angle = 90° before rotation after rotation (d) Enlargement: Elongated Spring Spring Load Spring 1 kg Load Compressed spring 232 Allied Mathematics-7 Thirdly, there is a rotation about a fixed point. Here also, there is no change in shape and size. The last one is different from the first three. Here, the shape remains the same but the size is changed, i.e., becomes large or small. These four types of changes or transformations are given the names: (a) Translation (b) Reflection (c) Rotation, and (d) Enlargement or Reduction. We give below some more diagrams to make the above terms more familiar. (a) Translation: after translation R R before translation (b) Reflection: SATURDAY FEBRUARY 15 SATURDAY FEBRUARY 15 Mirror R R before reflection mirror line after reflection (c) Rotation: R R angle = 90° angle = 90° before rotation after rotation (d) Enlargement: Elongated Spring Spring Load Spring 1 kg Load Compressed spring Sliding window Sliding window
Geometry GEOMETRY 299 CLASSWORK EXAMPLES Example 1 Figures in a plane can be reflected, rotated, or slid to produce new figures. When this is done, the new figure is called: (a) the pre-image (b) the image (c) an isometry (d) a dilation Solution: (b) the image Example 2 The figure on the left is the pre-image of the figure on the right (image). Name the type of transformation. (a) translation (b) reflection (c) rotation (d) none of the above Solution: (a) translation Example 3 The figure on the left is the pre-image of the figure on the right (image). Name the type of transformation. (a) translation (b) reflection (c) rotation (d) None-of above Solution: rotation. EXERCISE 19.1 Your mastery depends on practice. Practice like you play. 1. Tick mark the correct answer. (a) New figures in a plane can be produced by reflection, rotation, translation or translation. When this is done, the given figure is called: i. the pre image ii. the image iii. an isometry iv. a dilation (b) A magnification always maps a figure onto: i. its mirror image ii. a congruent figure iii. a similar figure iv. none of the above 2. The figure on the left is the pre-image of the figure on the right (image). Name the type of transformation: (a) i. translation ii. reflection iii. rotation iv. none of the above + + + +
300 The Leading Mathematics - 7 (b) i. translation ii. reflection iii. rotation iv. none of the above (c) i. translation ii. reflection iii. rotation iv. none of the above (d) i. translation ii. reflection iii. rotation iv. none of the above (e) i. translation ii. reflection iii. rotation iv. none of the above (f) i. translation ii. reflection iii. rotation iv. none of the above (g) i. translation ii. reflection iii. rotation iv. none of the above ANSWERS Consult with your teacher.