Page 1 We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about alongwith how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used. S.No. Name and figure Description Use 1. The Ruler A ruler ideally has no To draw line [or the straight markings on it. However, segments and edge] the ruler in your instruments to measure box is graduated into their lengths. centimetres along one edge (and sometimes into inches along the other edge). 2. The Compasses A pair – a pointer on one To mark off end and a pencil on the equal lengths other. but not to measure them. To draw arcs and circles. Pencil Pointer 14.1 Introduction Chapter 14 P P Pr r ra a ac c ct t ti i ic c ca a al l l G G Ge e eo o om m me e et t tr r ry y y Page 2 We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about alongwith how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used. S.No. Name and figure Description Use 1. The Ruler A ruler ideally has no To draw line [or the straight markings on it. However, segments and edge] the ruler in your instruments to measure box is graduated into their lengths. centimetres along one edge (and sometimes into inches along the other edge). 2. The Compasses A pair – a pointer on one To mark off end and a pencil on the equal lengths other. but not to measure them. To draw arcs and circles. Pencil Pointer 14.1 Introduction Chapter 14 P P Pr r ra a ac c ct t ti i ic c ca a al l l G G Ge e eo o om m me e et t tr r ry y y PRACTICAL GEOMETRY 275 3. The Divider A pair of pointers To compare lengths. 4. Set-Squares Two triangular To draw pieces – one of them perpendicular has 45°, 45°, 90° and parallel angles at the vertices lines. and the other has 30°, 60°, 90° angles at the vertices. 5. The Protractor A semi-circular To draw device graduated into and measure angles. 180 degree-parts. The measure starts from 0° on the right hand side and ends with 180° on the left hand side and vice-versa. We are going to consider “Ruler and compasses constructions”, using ruler, only to draw lines, and compasses, only to draw arcs. Be careful while doing these constructions. Here are some tips to help you. (a) Draw thin lines and mark points lightly. (b) Maintain instruments with sharp tips and fine edges. (c) Have two pencils in the box, one for insertion into the compasses and the other to draw lines or curves and mark points. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 9 8 7 6 5 4 3 2 1 Page 3 We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about alongwith how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used. S.No. Name and figure Description Use 1. The Ruler A ruler ideally has no To draw line [or the straight markings on it. However, segments and edge] the ruler in your instruments to measure box is graduated into their lengths. centimetres along one edge (and sometimes into inches along the other edge). 2. The Compasses A pair – a pointer on one To mark off end and a pencil on the equal lengths other. but not to measure them. To draw arcs and circles. Pencil Pointer 14.1 Introduction Chapter 14 P P Pr r ra a ac c ct t ti i ic c ca a al l l G G Ge e eo o om m me e et t tr r ry y y PRACTICAL GEOMETRY 275 3. The Divider A pair of pointers To compare lengths. 4. Set-Squares Two triangular To draw pieces – one of them perpendicular has 45°, 45°, 90° and parallel angles at the vertices lines. and the other has 30°, 60°, 90° angles at the vertices. 5. The Protractor A semi-circular To draw device graduated into and measure angles. 180 degree-parts. The measure starts from 0° on the right hand side and ends with 180° on the left hand side and vice-versa. We are going to consider “Ruler and compasses constructions”, using ruler, only to draw lines, and compasses, only to draw arcs. Be careful while doing these constructions. Here are some tips to help you. (a) Draw thin lines and mark points lightly. (b) Maintain instruments with sharp tips and fine edges. (c) Have two pencils in the box, one for insertion into the compasses and the other to draw lines or curves and mark points. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 9 8 7 6 5 4 3 2 1 MATHEMATICS 276 14.2 The Circle Look at the wheel shown here. Every point on its boundary is at an equal distance from its centre. Can you mention a few such objects and draw them? Think about five such objects which have this shape. 14.2.1 Construction of a circle when its radius is known Suppose we want to draw a circle of radius 3 cm. We need to use our compasses. Here are the steps to follow. Step 1 Open the compasses for the required radius of 3cm. Step 2 Mark a point with a sharp pencil where we want the centre of the circle to be. Name it as O. Step 3 Place the pointer of the compasses on O. Step 4 Turn the compasses slowly to draw the circle. Be careful to complete the movement around in one instant. Think, discuss and write How many circles can you draw with a given centre O and a point, say P? EXERCISE 14.1 1. Draw a circle of radius 3.2 cm. 2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm. 3. Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer? 4. Draw any circle and mark points A, B and C such that (a) A is on the circle. (b) B is in the interior of the circle. (c) C is in the exterior of the circle. 5. Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at C and D. Examine whether AB and CD are at right angles. Page 4 We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about alongwith how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used. S.No. Name and figure Description Use 1. The Ruler A ruler ideally has no To draw line [or the straight markings on it. However, segments and edge] the ruler in your instruments to measure box is graduated into their lengths. centimetres along one edge (and sometimes into inches along the other edge). 2. The Compasses A pair – a pointer on one To mark off end and a pencil on the equal lengths other. but not to measure them. To draw arcs and circles. Pencil Pointer 14.1 Introduction Chapter 14 P P Pr r ra a ac c ct t ti i ic c ca a al l l G G Ge e eo o om m me e et t tr r ry y y PRACTICAL GEOMETRY 275 3. The Divider A pair of pointers To compare lengths. 4. Set-Squares Two triangular To draw pieces – one of them perpendicular has 45°, 45°, 90° and parallel angles at the vertices lines. and the other has 30°, 60°, 90° angles at the vertices. 5. The Protractor A semi-circular To draw device graduated into and measure angles. 180 degree-parts. The measure starts from 0° on the right hand side and ends with 180° on the left hand side and vice-versa. We are going to consider “Ruler and compasses constructions”, using ruler, only to draw lines, and compasses, only to draw arcs. Be careful while doing these constructions. Here are some tips to help you. (a) Draw thin lines and mark points lightly. (b) Maintain instruments with sharp tips and fine edges. (c) Have two pencils in the box, one for insertion into the compasses and the other to draw lines or curves and mark points. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 9 8 7 6 5 4 3 2 1 MATHEMATICS 276 14.2 The Circle Look at the wheel shown here. Every point on its boundary is at an equal distance from its centre. Can you mention a few such objects and draw them? Think about five such objects which have this shape. 14.2.1 Construction of a circle when its radius is known Suppose we want to draw a circle of radius 3 cm. We need to use our compasses. Here are the steps to follow. Step 1 Open the compasses for the required radius of 3cm. Step 2 Mark a point with a sharp pencil where we want the centre of the circle to be. Name it as O. Step 3 Place the pointer of the compasses on O. Step 4 Turn the compasses slowly to draw the circle. Be careful to complete the movement around in one instant. Think, discuss and write How many circles can you draw with a given centre O and a point, say P? EXERCISE 14.1 1. Draw a circle of radius 3.2 cm. 2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm. 3. Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer? 4. Draw any circle and mark points A, B and C such that (a) A is on the circle. (b) B is in the interior of the circle. (c) C is in the exterior of the circle. 5. Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at C and D. Examine whether AB and CD are at right angles. PRACTICAL GEOMETRY 277 14.3 A Line Segment Remember that a line segment is bounded by two end-points. This makes it possible to measure its length with a ruler. If we know the length of a line segment, it becomes possible to represent it by a diagram. Let us see how we do this. 14.3.1 Construction of a line segment of a given length Suppose we want to draw a line segment of length 4.7 cm. We can use our ruler and mark two points A and B which are 4.7 cm apart. Join A and B and get AB . While marking the points A and B, we should look straight down at the measuring device. Otherwise we will get an incorrect value. Use of ruler and compasses A better method would be to use compasses to construct a line segment of a given length. Step 1 Draw a line l. Mark a point A on a line l. Step 2 Place the compasses pointer on the zero mark of the ruler. Open it to place the pencil point upto the 4.7cm mark. Step 3 Taking caution that the opening of the compasses has not changed, place the pointer on A and swing an arc to cut l at B. Step 4 AB is a line segment of required length. l Page 5 We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about alongwith how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used. S.No. Name and figure Description Use 1. The Ruler A ruler ideally has no To draw line [or the straight markings on it. However, segments and edge] the ruler in your instruments to measure box is graduated into their lengths. centimetres along one edge (and sometimes into inches along the other edge). 2. The Compasses A pair – a pointer on one To mark off end and a pencil on the equal lengths other. but not to measure them. To draw arcs and circles. Pencil Pointer 14.1 Introduction Chapter 14 P P Pr r ra a ac c ct t ti i ic c ca a al l l G G Ge e eo o om m me e et t tr r ry y y PRACTICAL GEOMETRY 275 3. The Divider A pair of pointers To compare lengths. 4. Set-Squares Two triangular To draw pieces – one of them perpendicular has 45°, 45°, 90° and parallel angles at the vertices lines. and the other has 30°, 60°, 90° angles at the vertices. 5. The Protractor A semi-circular To draw device graduated into and measure angles. 180 degree-parts. The measure starts from 0° on the right hand side and ends with 180° on the left hand side and vice-versa. We are going to consider “Ruler and compasses constructions”, using ruler, only to draw lines, and compasses, only to draw arcs. Be careful while doing these constructions. Here are some tips to help you. (a) Draw thin lines and mark points lightly. (b) Maintain instruments with sharp tips and fine edges. (c) Have two pencils in the box, one for insertion into the compasses and the other to draw lines or curves and mark points. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 9 8 7 6 5 4 3 2 1 MATHEMATICS 276 14.2 The Circle Look at the wheel shown here. Every point on its boundary is at an equal distance from its centre. Can you mention a few such objects and draw them? Think about five such objects which have this shape. 14.2.1 Construction of a circle when its radius is known Suppose we want to draw a circle of radius 3 cm. We need to use our compasses. Here are the steps to follow. Step 1 Open the compasses for the required radius of 3cm. Step 2 Mark a point with a sharp pencil where we want the centre of the circle to be. Name it as O. Step 3 Place the pointer of the compasses on O. Step 4 Turn the compasses slowly to draw the circle. Be careful to complete the movement around in one instant. Think, discuss and write How many circles can you draw with a given centre O and a point, say P? EXERCISE 14.1 1. Draw a circle of radius 3.2 cm. 2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm. 3. Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer? 4. Draw any circle and mark points A, B and C such that (a) A is on the circle. (b) B is in the interior of the circle. (c) C is in the exterior of the circle. 5. Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at C and D. Examine whether AB and CD are at right angles. PRACTICAL GEOMETRY 277 14.3 A Line Segment Remember that a line segment is bounded by two end-points. This makes it possible to measure its length with a ruler. If we know the length of a line segment, it becomes possible to represent it by a diagram. Let us see how we do this. 14.3.1 Construction of a line segment of a given length Suppose we want to draw a line segment of length 4.7 cm. We can use our ruler and mark two points A and B which are 4.7 cm apart. Join A and B and get AB . While marking the points A and B, we should look straight down at the measuring device. Otherwise we will get an incorrect value. Use of ruler and compasses A better method would be to use compasses to construct a line segment of a given length. Step 1 Draw a line l. Mark a point A on a line l. Step 2 Place the compasses pointer on the zero mark of the ruler. Open it to place the pencil point upto the 4.7cm mark. Step 3 Taking caution that the opening of the compasses has not changed, place the pointer on A and swing an arc to cut l at B. Step 4 AB is a line segment of required length. l MATHEMATICS 278 EXERCISE 14.2 1. Draw a line segment of length 7.3 cm using a ruler. 2. Construct a line segment of length 5.6 cm using ruler and compasses. 3. Construct AB of length 7.8 cm. From this, cut off AC of length 4.7 cm. Measure BC . 4. Given AB of length 3.9 cm, construct PQ such that the length of PQ is twice that of AB . Verify by measurement. (Hint : Construct PX such that length of PX = length of AB ; then cut off XQ such that XQ also has the length of AB .) 5. Given AB of length 7.3 cm and CD of length 3.4 cm, construct a line segment XY such that the length of XY is equal to the difference between the lengths of AB and CD . Verify by measurement. 14.3.2 Constructing a copy of a given line segment Suppose you want to draw a line segment whose length is equal to that of a given line segment AB . A quick and natural approach is to use your ruler (which is marked with centimetres and millimetres) to measure the length of AB and then use the same length to draw another line segment CD . A second approach would be to use a transparent sheet and trace AB onto another portion of the paper. But these methods may not always give accurate results. A better approach would be to use ruler and compasses for making this construction. To make a copy of AB . Step 1 Given AB whose length is not known. A BRead More

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