Page 1 EXERCISE 14 A Q. 1. Draw a line segment PQ = 6·2 cm. Draw the perpendicular bisector of PQ. Sol. Steps of construction : (i) Draw a line segment PQ = 6·2 cm (ii) With centre P and Q and radius more than half of PQ, draw arcs on each side intersecting each other at L and M. (iii) Join LM intersecting PQ at N. Then, LM is the perpendicular bisector of PQ. Q. 2. Draw a line segment AB = 5.6 cm. Draw the perpendicular bisector of AB. Sol. Steps of Construction : 1. Draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one one each side of AB. 3. With B as centre and same radius as before, draw arcs, cutting the previous arcs at P and Q respectively. 4. Join P and Q, meeting AB at M. Then PQ is the required perpendicular bisector of AB. Verification : Measure AMP. We see that AMP = 90º. So, PQ is the perpendicular bisector of AB. Q. 3. Draw an angle equal to AOB given in the adjoining figure : Page 2 EXERCISE 14 A Q. 1. Draw a line segment PQ = 6·2 cm. Draw the perpendicular bisector of PQ. Sol. Steps of construction : (i) Draw a line segment PQ = 6·2 cm (ii) With centre P and Q and radius more than half of PQ, draw arcs on each side intersecting each other at L and M. (iii) Join LM intersecting PQ at N. Then, LM is the perpendicular bisector of PQ. Q. 2. Draw a line segment AB = 5.6 cm. Draw the perpendicular bisector of AB. Sol. Steps of Construction : 1. Draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one one each side of AB. 3. With B as centre and same radius as before, draw arcs, cutting the previous arcs at P and Q respectively. 4. Join P and Q, meeting AB at M. Then PQ is the required perpendicular bisector of AB. Verification : Measure AMP. We see that AMP = 90º. So, PQ is the perpendicular bisector of AB. Q. 3. Draw an angle equal to AOB given in the adjoining figure : Sol. Steps of Contruction : 1. Draw a ray RX. 2. With O as centre and any radius draw an arc cutting OA and OB at P and Q respectively. 3. With R as centre and same radius draw an arc cutting RX at S. 4. With S as centre and radius PQ cut the arc through S at T. 5. Join RT and produce it to Y. Then XRY is the required angle equal to AOB. Verification : Measuring angle AOB and XRY, we observe that XRY = AOB. Q. 4. Draw an angle of 50º with the help of a protractor. Draw a ray bisecting this angle. Sol. Steps of constructions : (i) Draw an angle ABC = 50º with the help of a protractor. (ii) With centre B and C and a suitable radius, draw an arc meeting AB at Q and BC at P. (iii) With centres P and Q and with a suitable radius draw two arcs intersecting each other at R inside the angle ABC. (iv) Join RB. Then ray BR is the bisector of ABC. Q. 5. Construct AOB = 85º with the help of a protractor. Draw a ray OX bisecting AOB. Sol. Steps of construction : (i) Draw an angle AOB = 85º with the help of the protractor. (ii) With centre O, draw an arc with a suitable radius meeting OB at E and OA at F. (iii) With centre E and F and with a suitable radius draw arcs intersecting each other at X inside the angle AOB. Then ray OX is the bisector of AOB. Q. 6. Draw a line AB. Take a point P on it. Draw a line passing through P and perpendicular to AB. Page 3 EXERCISE 14 A Q. 1. Draw a line segment PQ = 6·2 cm. Draw the perpendicular bisector of PQ. Sol. Steps of construction : (i) Draw a line segment PQ = 6·2 cm (ii) With centre P and Q and radius more than half of PQ, draw arcs on each side intersecting each other at L and M. (iii) Join LM intersecting PQ at N. Then, LM is the perpendicular bisector of PQ. Q. 2. Draw a line segment AB = 5.6 cm. Draw the perpendicular bisector of AB. Sol. Steps of Construction : 1. Draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one one each side of AB. 3. With B as centre and same radius as before, draw arcs, cutting the previous arcs at P and Q respectively. 4. Join P and Q, meeting AB at M. Then PQ is the required perpendicular bisector of AB. Verification : Measure AMP. We see that AMP = 90º. So, PQ is the perpendicular bisector of AB. Q. 3. Draw an angle equal to AOB given in the adjoining figure : Sol. Steps of Contruction : 1. Draw a ray RX. 2. With O as centre and any radius draw an arc cutting OA and OB at P and Q respectively. 3. With R as centre and same radius draw an arc cutting RX at S. 4. With S as centre and radius PQ cut the arc through S at T. 5. Join RT and produce it to Y. Then XRY is the required angle equal to AOB. Verification : Measuring angle AOB and XRY, we observe that XRY = AOB. Q. 4. Draw an angle of 50º with the help of a protractor. Draw a ray bisecting this angle. Sol. Steps of constructions : (i) Draw an angle ABC = 50º with the help of a protractor. (ii) With centre B and C and a suitable radius, draw an arc meeting AB at Q and BC at P. (iii) With centres P and Q and with a suitable radius draw two arcs intersecting each other at R inside the angle ABC. (iv) Join RB. Then ray BR is the bisector of ABC. Q. 5. Construct AOB = 85º with the help of a protractor. Draw a ray OX bisecting AOB. Sol. Steps of construction : (i) Draw an angle AOB = 85º with the help of the protractor. (ii) With centre O, draw an arc with a suitable radius meeting OB at E and OA at F. (iii) With centre E and F and with a suitable radius draw arcs intersecting each other at X inside the angle AOB. Then ray OX is the bisector of AOB. Q. 6. Draw a line AB. Take a point P on it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P on it. 2. With P as centre and any suitable radius draw a semi-circle to cut the line AB at X and Y. 3. With centre X and radius more than XP draw an arc. 4. With centre Y and same radius draw another arc to cut the previous arc at Q. 5. Join PQ. Then, PQ is the required line passing through P and perpendicular to AB. Verification : Measure APQ, we see that APQ = 90º Q. 7. Draw a line AB. Take a point P outside it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P outside it. 2. With P as centre and suitable radius, draw an arc intersecting AB at C and D. 3. With C as centre and radius more than half CD, draw an arc. 4. With D as centre and same radius, draw another arc to cut the previous arc at Q. 5. Join PQ, meeting AB at L. Then PL is the required line passing thrugh P and perpendicular to AB. Verification : Measure PLB. We see that PLB = 90º. Q. 8. Draw a line AB. Take a point P outside it. Draw a line passing through P and parallel to AB. Sol. Steps of Construction : 1. Draw a given line AB and take a point P outside it. 2. Take a point R on AB. 3. Join PR. 4. Draw RPQ such that RPQ = PRB as shown in the figure. 5. Produce PQ on both sides to form a line. Then, PQ is the required line passing through P and parallel to AB. Verification : Since RPQ = PRB and these are alternate interior angles, it follows that PQ | | AB. Q. 9. Draw an BAC of measure 60º such that AB = 4.5 cm and AC = 5 cm. Through C draw a line parallel to AB and through B draw a line parallel to AC, intersecting each other at D. Measure BD and CD. Sol. Steps of Construction : 1. Draw a ray AX and cut of AC = 5 cm. 2. With A as centre and suitable radius draw an arc above AX and cutting it at P. Page 4 EXERCISE 14 A Q. 1. Draw a line segment PQ = 6·2 cm. Draw the perpendicular bisector of PQ. Sol. Steps of construction : (i) Draw a line segment PQ = 6·2 cm (ii) With centre P and Q and radius more than half of PQ, draw arcs on each side intersecting each other at L and M. (iii) Join LM intersecting PQ at N. Then, LM is the perpendicular bisector of PQ. Q. 2. Draw a line segment AB = 5.6 cm. Draw the perpendicular bisector of AB. Sol. Steps of Construction : 1. Draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one one each side of AB. 3. With B as centre and same radius as before, draw arcs, cutting the previous arcs at P and Q respectively. 4. Join P and Q, meeting AB at M. Then PQ is the required perpendicular bisector of AB. Verification : Measure AMP. We see that AMP = 90º. So, PQ is the perpendicular bisector of AB. Q. 3. Draw an angle equal to AOB given in the adjoining figure : Sol. Steps of Contruction : 1. Draw a ray RX. 2. With O as centre and any radius draw an arc cutting OA and OB at P and Q respectively. 3. With R as centre and same radius draw an arc cutting RX at S. 4. With S as centre and radius PQ cut the arc through S at T. 5. Join RT and produce it to Y. Then XRY is the required angle equal to AOB. Verification : Measuring angle AOB and XRY, we observe that XRY = AOB. Q. 4. Draw an angle of 50º with the help of a protractor. Draw a ray bisecting this angle. Sol. Steps of constructions : (i) Draw an angle ABC = 50º with the help of a protractor. (ii) With centre B and C and a suitable radius, draw an arc meeting AB at Q and BC at P. (iii) With centres P and Q and with a suitable radius draw two arcs intersecting each other at R inside the angle ABC. (iv) Join RB. Then ray BR is the bisector of ABC. Q. 5. Construct AOB = 85º with the help of a protractor. Draw a ray OX bisecting AOB. Sol. Steps of construction : (i) Draw an angle AOB = 85º with the help of the protractor. (ii) With centre O, draw an arc with a suitable radius meeting OB at E and OA at F. (iii) With centre E and F and with a suitable radius draw arcs intersecting each other at X inside the angle AOB. Then ray OX is the bisector of AOB. Q. 6. Draw a line AB. Take a point P on it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P on it. 2. With P as centre and any suitable radius draw a semi-circle to cut the line AB at X and Y. 3. With centre X and radius more than XP draw an arc. 4. With centre Y and same radius draw another arc to cut the previous arc at Q. 5. Join PQ. Then, PQ is the required line passing through P and perpendicular to AB. Verification : Measure APQ, we see that APQ = 90º Q. 7. Draw a line AB. Take a point P outside it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P outside it. 2. With P as centre and suitable radius, draw an arc intersecting AB at C and D. 3. With C as centre and radius more than half CD, draw an arc. 4. With D as centre and same radius, draw another arc to cut the previous arc at Q. 5. Join PQ, meeting AB at L. Then PL is the required line passing thrugh P and perpendicular to AB. Verification : Measure PLB. We see that PLB = 90º. Q. 8. Draw a line AB. Take a point P outside it. Draw a line passing through P and parallel to AB. Sol. Steps of Construction : 1. Draw a given line AB and take a point P outside it. 2. Take a point R on AB. 3. Join PR. 4. Draw RPQ such that RPQ = PRB as shown in the figure. 5. Produce PQ on both sides to form a line. Then, PQ is the required line passing through P and parallel to AB. Verification : Since RPQ = PRB and these are alternate interior angles, it follows that PQ | | AB. Q. 9. Draw an BAC of measure 60º such that AB = 4.5 cm and AC = 5 cm. Through C draw a line parallel to AB and through B draw a line parallel to AC, intersecting each other at D. Measure BD and CD. Sol. Steps of Construction : 1. Draw a ray AX and cut of AC = 5 cm. 2. With A as centre and suitable radius draw an arc above AX and cutting it at P. 3. With P as centre and the same radius as before draw another arc to cut the previous arc at Q. 4. Join PQ and produce it to the point B such that. AB = 4.5 cm. Then BAC = 60º is the required angle. 5. Draw ÐRBA such that ÐRBA = ÐBAC. 6. Produce RB on both sides to form a line. Then, RY is the line parallel to AC and passing through B. 7. Now, draw ÐSCX = ÐBAC at the point C. 8. Produce CS to intersect the line RY at D. Then CD is the required line thrugh C and parallel to AB. 9. Measure BD and CD. We see that BD = 5 cm. and CD = 4.5 cm. Verification. Since RBA = BAC and these are alternate angles, it follows that RY | | AC. Also SCX = BAC and these are corresponding angles, it follows that CD | | AB. Q. 10. Draw a line segment AB = 6 cm. Take a point C on AB such that AC = 2.5 cm. Draw CD perpendicular to AB. Sol. Steps of Construction : 1. With the help of a rular, draw a line segment AB = 6 cm. and off AC = 2.5 cm such that the point C is on AB. 2. With C as centre and any suitable radius draw a semi-circle to cut AB at P and Q. 3. With P as centre and any radius more than PC draw an arc. 4. With Q as centre and same radius draw another arc to cut the previous arc at D. 5. Join CD. Then CD is the required line perpendicular to AB. Verification : Measure ACD. We see that ACD = 90º. Q. 11. Draw a line segment AB = 5.6 cm. Draw the right bisector of AB. Sol. Steps of Construction : 1. With the help of rular, draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one on each side of AB. 3. With B as centre and the same radius as before draw arcs, cutting the previous arcs at P and Q respectively. 4. Join PQ, meeting AB at M. Then, PQ is the required right bisector of AB. Verification : On measuring AM and BM and AMP, we see that AM = BM and AMP = 90º. So, PQ is the right bisector of AB. Q. 12. Using a rular and a pair of compasses, construct AOB = 60º and draw the bisector of this angle. Sol. Steps of Construction : 1. With the help of a rular, draw a ray OA. 2. With O as centre and suitable radius draw an arc to cut OA at P. Page 5 EXERCISE 14 A Q. 1. Draw a line segment PQ = 6·2 cm. Draw the perpendicular bisector of PQ. Sol. Steps of construction : (i) Draw a line segment PQ = 6·2 cm (ii) With centre P and Q and radius more than half of PQ, draw arcs on each side intersecting each other at L and M. (iii) Join LM intersecting PQ at N. Then, LM is the perpendicular bisector of PQ. Q. 2. Draw a line segment AB = 5.6 cm. Draw the perpendicular bisector of AB. Sol. Steps of Construction : 1. Draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one one each side of AB. 3. With B as centre and same radius as before, draw arcs, cutting the previous arcs at P and Q respectively. 4. Join P and Q, meeting AB at M. Then PQ is the required perpendicular bisector of AB. Verification : Measure AMP. We see that AMP = 90º. So, PQ is the perpendicular bisector of AB. Q. 3. Draw an angle equal to AOB given in the adjoining figure : Sol. Steps of Contruction : 1. Draw a ray RX. 2. With O as centre and any radius draw an arc cutting OA and OB at P and Q respectively. 3. With R as centre and same radius draw an arc cutting RX at S. 4. With S as centre and radius PQ cut the arc through S at T. 5. Join RT and produce it to Y. Then XRY is the required angle equal to AOB. Verification : Measuring angle AOB and XRY, we observe that XRY = AOB. Q. 4. Draw an angle of 50º with the help of a protractor. Draw a ray bisecting this angle. Sol. Steps of constructions : (i) Draw an angle ABC = 50º with the help of a protractor. (ii) With centre B and C and a suitable radius, draw an arc meeting AB at Q and BC at P. (iii) With centres P and Q and with a suitable radius draw two arcs intersecting each other at R inside the angle ABC. (iv) Join RB. Then ray BR is the bisector of ABC. Q. 5. Construct AOB = 85º with the help of a protractor. Draw a ray OX bisecting AOB. Sol. Steps of construction : (i) Draw an angle AOB = 85º with the help of the protractor. (ii) With centre O, draw an arc with a suitable radius meeting OB at E and OA at F. (iii) With centre E and F and with a suitable radius draw arcs intersecting each other at X inside the angle AOB. Then ray OX is the bisector of AOB. Q. 6. Draw a line AB. Take a point P on it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P on it. 2. With P as centre and any suitable radius draw a semi-circle to cut the line AB at X and Y. 3. With centre X and radius more than XP draw an arc. 4. With centre Y and same radius draw another arc to cut the previous arc at Q. 5. Join PQ. Then, PQ is the required line passing through P and perpendicular to AB. Verification : Measure APQ, we see that APQ = 90º Q. 7. Draw a line AB. Take a point P outside it. Draw a line passing through P and perpendicular to AB. Sol. Steps of Construction : 1. Draw the given line AB and take a point P outside it. 2. With P as centre and suitable radius, draw an arc intersecting AB at C and D. 3. With C as centre and radius more than half CD, draw an arc. 4. With D as centre and same radius, draw another arc to cut the previous arc at Q. 5. Join PQ, meeting AB at L. Then PL is the required line passing thrugh P and perpendicular to AB. Verification : Measure PLB. We see that PLB = 90º. Q. 8. Draw a line AB. Take a point P outside it. Draw a line passing through P and parallel to AB. Sol. Steps of Construction : 1. Draw a given line AB and take a point P outside it. 2. Take a point R on AB. 3. Join PR. 4. Draw RPQ such that RPQ = PRB as shown in the figure. 5. Produce PQ on both sides to form a line. Then, PQ is the required line passing through P and parallel to AB. Verification : Since RPQ = PRB and these are alternate interior angles, it follows that PQ | | AB. Q. 9. Draw an BAC of measure 60º such that AB = 4.5 cm and AC = 5 cm. Through C draw a line parallel to AB and through B draw a line parallel to AC, intersecting each other at D. Measure BD and CD. Sol. Steps of Construction : 1. Draw a ray AX and cut of AC = 5 cm. 2. With A as centre and suitable radius draw an arc above AX and cutting it at P. 3. With P as centre and the same radius as before draw another arc to cut the previous arc at Q. 4. Join PQ and produce it to the point B such that. AB = 4.5 cm. Then BAC = 60º is the required angle. 5. Draw ÐRBA such that ÐRBA = ÐBAC. 6. Produce RB on both sides to form a line. Then, RY is the line parallel to AC and passing through B. 7. Now, draw ÐSCX = ÐBAC at the point C. 8. Produce CS to intersect the line RY at D. Then CD is the required line thrugh C and parallel to AB. 9. Measure BD and CD. We see that BD = 5 cm. and CD = 4.5 cm. Verification. Since RBA = BAC and these are alternate angles, it follows that RY | | AC. Also SCX = BAC and these are corresponding angles, it follows that CD | | AB. Q. 10. Draw a line segment AB = 6 cm. Take a point C on AB such that AC = 2.5 cm. Draw CD perpendicular to AB. Sol. Steps of Construction : 1. With the help of a rular, draw a line segment AB = 6 cm. and off AC = 2.5 cm such that the point C is on AB. 2. With C as centre and any suitable radius draw a semi-circle to cut AB at P and Q. 3. With P as centre and any radius more than PC draw an arc. 4. With Q as centre and same radius draw another arc to cut the previous arc at D. 5. Join CD. Then CD is the required line perpendicular to AB. Verification : Measure ACD. We see that ACD = 90º. Q. 11. Draw a line segment AB = 5.6 cm. Draw the right bisector of AB. Sol. Steps of Construction : 1. With the help of rular, draw a line segment AB = 5.6 cm. 2. With A as centre and radius more than half AB, draw arcs, one on each side of AB. 3. With B as centre and the same radius as before draw arcs, cutting the previous arcs at P and Q respectively. 4. Join PQ, meeting AB at M. Then, PQ is the required right bisector of AB. Verification : On measuring AM and BM and AMP, we see that AM = BM and AMP = 90º. So, PQ is the right bisector of AB. Q. 12. Using a rular and a pair of compasses, construct AOB = 60º and draw the bisector of this angle. Sol. Steps of Construction : 1. With the help of a rular, draw a ray OA. 2. With O as centre and suitable radius draw an arc to cut OA at P. 3. With P as centre and the same radius, draw another are to cut the previous arc at Q. 4. Join OQ and produce it to any point B, then AOB = 60º is the required angle. 5. With P as centre and radius more than half PQ, draw an arc. 6. With Q as centre and the same radius, draw another arc to cut the previous arc at R. 7. Join OR and produce it to the point C. Then OC is the required bisector of AOB. Verification : Measure AOC and BOC. We see that AOC = BOC. So, OC is the bisector of AOB. Q. 13. Construct an angle of 135º, using a rular and pair of compasses. Sol. Steps of construction : 1. Draw a ray OA with the help of a rular. 2. With O as centre and suitable radius draw an arc above OA to cut it at P . 3. With P as centre and same radius, cut the arc at Q and again with Q as centre and same radius cut the arc at R. With R as centre and same radius, again cut the arc at S. 4. Join OR and produce it to B and join OS and produce it to C. 5. Draw the bisector OD of BOC. 6. Draw the bisector OE of BOD. Then, AOE = 135º is the required angle. EXERCISE 14B Q. 1. Using a pair of compasses construct the following angles : (i) 60º (ii) 120º (iii) 90º Sol. (i) 60º Steps of construction : (i) Draw a ray OA. (ii) With centre O and with a suitable radius drawn an arc meeting OA at E. (iii) With centre E and with same radius, draw another arc cutting the first arc at F. (iv) Join OF and produce it to B Then AOB = 60º (ii) 120º Steps of construction : (i) Draw a ray OA (ii) With centre O and with a suitable radius draw an arc meeting OA at E.Read More

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