Page 1 PRACTICAL GEOMETRY 57 DO THIS 4.1 Introduction Y ou have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely , a quadrilateral. T ake a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm. This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle (Fig 4.2)? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same. Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same. This shows that 4 measurements of a quadrilateral cannot determine it uniquely . Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity! Practical Geometry CHAPTER 4 Fig 4.1 Fig 4.2 Fig 4.3 Page 2 PRACTICAL GEOMETRY 57 DO THIS 4.1 Introduction Y ou have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely , a quadrilateral. T ake a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm. This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle (Fig 4.2)? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same. Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same. This shows that 4 measurements of a quadrilateral cannot determine it uniquely . Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity! Practical Geometry CHAPTER 4 Fig 4.1 Fig 4.2 Fig 4.3 58 MATHEMATICS THINK, DISCUSS AND WRITE Y ou have constructed a rectangle with two sticks each of length 10 cm and other two sticks each of length 8 cm. Now introduce another stick of length equal to BD and tie it along BD (Fig 4.4). If you push the breadth now, does the shape change? No! It cannot, without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely , i.e., there is no other quadrilateral (with the given lengths of sides) possible now . Thus, we observe that five measurements can determine a quadrilateral uniquely . But will any five measurements (of sides and angles) be sufficient to draw a unique quadrilateral? Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, ?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer. 4.2 Constructing a Quadrilateral We shall learn how to construct a unique quadrilateral given the following measurements: â€¢ When four sides and one diagonal are given. â€¢ When two diagonals and three sides are given. â€¢ When two adjacent sides and three angles are given. â€¢ When three sides and two included angles are given. â€¢ When other special properties are known. Let us take up these constructions one-by-one. 4.2.1 When the lengths of four sides and a diagonal are given W e shall explain this construction through an example. Example 1: Construct a quadrilateral PQRS where PQ = 4 cm,QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. Solution: [A rough sketch will help us in visualising the quadrilateral. W e draw this first and mark the measurements.] (Fig 4.5) Fig 4.4 6 cm 5.5 cm 5 cm 4 cm Q PR S 7 cm Fig 4.5 Page 3 PRACTICAL GEOMETRY 57 DO THIS 4.1 Introduction Y ou have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely , a quadrilateral. T ake a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm. This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle (Fig 4.2)? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same. Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same. This shows that 4 measurements of a quadrilateral cannot determine it uniquely . Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity! Practical Geometry CHAPTER 4 Fig 4.1 Fig 4.2 Fig 4.3 58 MATHEMATICS THINK, DISCUSS AND WRITE Y ou have constructed a rectangle with two sticks each of length 10 cm and other two sticks each of length 8 cm. Now introduce another stick of length equal to BD and tie it along BD (Fig 4.4). If you push the breadth now, does the shape change? No! It cannot, without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely , i.e., there is no other quadrilateral (with the given lengths of sides) possible now . Thus, we observe that five measurements can determine a quadrilateral uniquely . But will any five measurements (of sides and angles) be sufficient to draw a unique quadrilateral? Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, ?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer. 4.2 Constructing a Quadrilateral We shall learn how to construct a unique quadrilateral given the following measurements: â€¢ When four sides and one diagonal are given. â€¢ When two diagonals and three sides are given. â€¢ When two adjacent sides and three angles are given. â€¢ When three sides and two included angles are given. â€¢ When other special properties are known. Let us take up these constructions one-by-one. 4.2.1 When the lengths of four sides and a diagonal are given W e shall explain this construction through an example. Example 1: Construct a quadrilateral PQRS where PQ = 4 cm,QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. Solution: [A rough sketch will help us in visualising the quadrilateral. W e draw this first and mark the measurements.] (Fig 4.5) Fig 4.4 6 cm 5.5 cm 5 cm 4 cm Q PR S 7 cm Fig 4.5 PRACTICAL GEOMETRY 59 Step 1 From the rough sketch, it is easy to see that ?PQR can be constructed using SSS construction condition. Draw ?PQR (Fig 4.6). Step 2 Now, we have to locate the fourth point S. This â€˜Sâ€™ would be on the side opposite to Q with reference to PR. For that, we have two measurements. S is 5.5 cm away from P . So, with P as centre, draw an arc of radius 5.5 cm. (The point S is somewhere on this arc!) (Fig 4.7). Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The point S is somewhere on this arc also!) (Fig 4.8). Fig 4.6 Fig 4.7 Fig 4.8 Page 4 PRACTICAL GEOMETRY 57 DO THIS 4.1 Introduction Y ou have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely , a quadrilateral. T ake a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm. This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle (Fig 4.2)? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same. Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same. This shows that 4 measurements of a quadrilateral cannot determine it uniquely . Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity! Practical Geometry CHAPTER 4 Fig 4.1 Fig 4.2 Fig 4.3 58 MATHEMATICS THINK, DISCUSS AND WRITE Y ou have constructed a rectangle with two sticks each of length 10 cm and other two sticks each of length 8 cm. Now introduce another stick of length equal to BD and tie it along BD (Fig 4.4). If you push the breadth now, does the shape change? No! It cannot, without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely , i.e., there is no other quadrilateral (with the given lengths of sides) possible now . Thus, we observe that five measurements can determine a quadrilateral uniquely . But will any five measurements (of sides and angles) be sufficient to draw a unique quadrilateral? Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, ?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer. 4.2 Constructing a Quadrilateral We shall learn how to construct a unique quadrilateral given the following measurements: â€¢ When four sides and one diagonal are given. â€¢ When two diagonals and three sides are given. â€¢ When two adjacent sides and three angles are given. â€¢ When three sides and two included angles are given. â€¢ When other special properties are known. Let us take up these constructions one-by-one. 4.2.1 When the lengths of four sides and a diagonal are given W e shall explain this construction through an example. Example 1: Construct a quadrilateral PQRS where PQ = 4 cm,QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. Solution: [A rough sketch will help us in visualising the quadrilateral. W e draw this first and mark the measurements.] (Fig 4.5) Fig 4.4 6 cm 5.5 cm 5 cm 4 cm Q PR S 7 cm Fig 4.5 PRACTICAL GEOMETRY 59 Step 1 From the rough sketch, it is easy to see that ?PQR can be constructed using SSS construction condition. Draw ?PQR (Fig 4.6). Step 2 Now, we have to locate the fourth point S. This â€˜Sâ€™ would be on the side opposite to Q with reference to PR. For that, we have two measurements. S is 5.5 cm away from P . So, with P as centre, draw an arc of radius 5.5 cm. (The point S is somewhere on this arc!) (Fig 4.7). Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The point S is somewhere on this arc also!) (Fig 4.8). Fig 4.6 Fig 4.7 Fig 4.8 60 MATHEMATICS THINK, DISCUSS AND WRITE Step 4 S should lie on both the arcs drawn. So it is the point of intersection of the two arcs. Mark S and complete PQRS. PQRS is the required quadrilateral (Fig 4.9). (i) We saw that 5 measurements of a quadrilateral can determine a quadrilateral uniquely . Do you think any five measurements of the quadrilateral can do this? (ii) Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why? (iii) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why? (iv) A student attempted to draw a quadrilateral PLA Y where PL = 3 cm, LA = 4 cm, AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it. What is the reason? [Hint: Discuss it using a rough sketch]. EXERCISE 4.1 1. Construct the following quadrilaterals. ( i ) Quadrilateral ABCD. (ii) Quadrilateral JUMP AB = 4.5 cm JU = 3.5 cm BC = 5.5 cm UM = 4 cm CD = 4 cm MP = 5 cm AD = 6 cm PJ = 4.5 cm AC = 7 cm PU = 6.5 cm (iii) Parallelogram MORE (iv) Rhombus BEST OR = 6 cm BE = 4.5 cm RE = 4.5 cm ET = 6 cm EO = 7.5 cm Fig 4.9 Page 5 PRACTICAL GEOMETRY 57 DO THIS 4.1 Introduction Y ou have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely , a quadrilateral. T ake a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm. This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle (Fig 4.2)? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same. Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same. This shows that 4 measurements of a quadrilateral cannot determine it uniquely . Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity! Practical Geometry CHAPTER 4 Fig 4.1 Fig 4.2 Fig 4.3 58 MATHEMATICS THINK, DISCUSS AND WRITE Y ou have constructed a rectangle with two sticks each of length 10 cm and other two sticks each of length 8 cm. Now introduce another stick of length equal to BD and tie it along BD (Fig 4.4). If you push the breadth now, does the shape change? No! It cannot, without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely , i.e., there is no other quadrilateral (with the given lengths of sides) possible now . Thus, we observe that five measurements can determine a quadrilateral uniquely . But will any five measurements (of sides and angles) be sufficient to draw a unique quadrilateral? Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, ?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer. 4.2 Constructing a Quadrilateral We shall learn how to construct a unique quadrilateral given the following measurements: â€¢ When four sides and one diagonal are given. â€¢ When two diagonals and three sides are given. â€¢ When two adjacent sides and three angles are given. â€¢ When three sides and two included angles are given. â€¢ When other special properties are known. Let us take up these constructions one-by-one. 4.2.1 When the lengths of four sides and a diagonal are given W e shall explain this construction through an example. Example 1: Construct a quadrilateral PQRS where PQ = 4 cm,QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. Solution: [A rough sketch will help us in visualising the quadrilateral. W e draw this first and mark the measurements.] (Fig 4.5) Fig 4.4 6 cm 5.5 cm 5 cm 4 cm Q PR S 7 cm Fig 4.5 PRACTICAL GEOMETRY 59 Step 1 From the rough sketch, it is easy to see that ?PQR can be constructed using SSS construction condition. Draw ?PQR (Fig 4.6). Step 2 Now, we have to locate the fourth point S. This â€˜Sâ€™ would be on the side opposite to Q with reference to PR. For that, we have two measurements. S is 5.5 cm away from P . So, with P as centre, draw an arc of radius 5.5 cm. (The point S is somewhere on this arc!) (Fig 4.7). Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The point S is somewhere on this arc also!) (Fig 4.8). Fig 4.6 Fig 4.7 Fig 4.8 60 MATHEMATICS THINK, DISCUSS AND WRITE Step 4 S should lie on both the arcs drawn. So it is the point of intersection of the two arcs. Mark S and complete PQRS. PQRS is the required quadrilateral (Fig 4.9). (i) We saw that 5 measurements of a quadrilateral can determine a quadrilateral uniquely . Do you think any five measurements of the quadrilateral can do this? (ii) Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why? (iii) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why? (iv) A student attempted to draw a quadrilateral PLA Y where PL = 3 cm, LA = 4 cm, AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it. What is the reason? [Hint: Discuss it using a rough sketch]. EXERCISE 4.1 1. Construct the following quadrilaterals. ( i ) Quadrilateral ABCD. (ii) Quadrilateral JUMP AB = 4.5 cm JU = 3.5 cm BC = 5.5 cm UM = 4 cm CD = 4 cm MP = 5 cm AD = 6 cm PJ = 4.5 cm AC = 7 cm PU = 6.5 cm (iii) Parallelogram MORE (iv) Rhombus BEST OR = 6 cm BE = 4.5 cm RE = 4.5 cm ET = 6 cm EO = 7.5 cm Fig 4.9 PRACTICAL GEOMETRY 61 Fig 4.12 4.2.2 When two diagonals and three sides are given When four sides and a diagonal were given, we first drew a triangle with the available data and then tried to locate the fourth point. The same technique is used here. Example 2: Construct a quadrilateral ABCD, given that BC = 4.5 cm, AD = 5.5 cm, CD = 5 cm the diagonal AC = 5.5 cm and diagonal BD = 7 cm. Solution: Here is the rough sketch of the quadrilateral ABCD (Fig 4.10). Studying this sketch, we can easily see that it is possible to draw ? ACD first (How?). Step 1 Draw ? ACD using SSS construction (Fig 4.11). (W e now need to find B at a distance of 4.5 cm from C and 7 cm from D). Step 2 With D as centre, draw an arc of radius 7 cm. (B is somewhere on this arc) (Fig 4.12). Step 3 With C as centre, draw an arc of radius 4.5 cm (B is somewhere on this arc also) (Fig 4.13). Fig 4.13 Fig 4.11 Fig 4.10Read More

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