Selina Textbook Solutions: Circle : Constructions | Mathematics Class 10 ICSE PDF Download

 Page 1


Constructions (Circles) 
 
Exercise 19 
Question 1. 
Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the 
circle drawn. Draw two tangents PA and PB to the given circle and measure the length 
of each tangent. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius 3 cm. 
2. From O, take a point P such that OP = 5 cm 
3. Draw a bisector of OP which intersects OP at M. 
4. With centre M, and radius OM, draw a circle which intersects the given circle at A 
and B. 
5. Join AP and BP. 
AP and BP are the required tangents. 
On measuring AP = BP = 4 cm 
Question 2. 
Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre 
of the circle. Draw tangents to the given circle from this exterior point. Measure the 
length of each tangent. 
Page 2


Constructions (Circles) 
 
Exercise 19 
Question 1. 
Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the 
circle drawn. Draw two tangents PA and PB to the given circle and measure the length 
of each tangent. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius 3 cm. 
2. From O, take a point P such that OP = 5 cm 
3. Draw a bisector of OP which intersects OP at M. 
4. With centre M, and radius OM, draw a circle which intersects the given circle at A 
and B. 
5. Join AP and BP. 
AP and BP are the required tangents. 
On measuring AP = BP = 4 cm 
Question 2. 
Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre 
of the circle. Draw tangents to the given circle from this exterior point. Measure the 
length of each tangent. 
Solution: 
 
1. Draw a circle of diameter 9 cm, taking O as the centre. 
2. Mark a point P outside the circle, such that PO = 7.5 cm. 
3. Taking OP as the diameter, draw a circle such that it cuts the earlier circle at A and 
B. 
4. Join PA and PB. 
Thus, PA and PB are required tangents. PA = PB = 6 cm 
Question 3. 
Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between 
the tangents is 45º. 
 
Solution: 
 
Steps of Construction: 
 
 
 
1. Draw a circle with centre O and radius BC = 5 cm 
2. Draw arcs making an angle of 180º- 45º = 135º at O such that ?AOB = 135º 
Page 3


Constructions (Circles) 
 
Exercise 19 
Question 1. 
Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the 
circle drawn. Draw two tangents PA and PB to the given circle and measure the length 
of each tangent. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius 3 cm. 
2. From O, take a point P such that OP = 5 cm 
3. Draw a bisector of OP which intersects OP at M. 
4. With centre M, and radius OM, draw a circle which intersects the given circle at A 
and B. 
5. Join AP and BP. 
AP and BP are the required tangents. 
On measuring AP = BP = 4 cm 
Question 2. 
Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre 
of the circle. Draw tangents to the given circle from this exterior point. Measure the 
length of each tangent. 
Solution: 
 
1. Draw a circle of diameter 9 cm, taking O as the centre. 
2. Mark a point P outside the circle, such that PO = 7.5 cm. 
3. Taking OP as the diameter, draw a circle such that it cuts the earlier circle at A and 
B. 
4. Join PA and PB. 
Thus, PA and PB are required tangents. PA = PB = 6 cm 
Question 3. 
Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between 
the tangents is 45º. 
 
Solution: 
 
Steps of Construction: 
 
 
 
1. Draw a circle with centre O and radius BC = 5 cm 
2. Draw arcs making an angle of 180º- 45º = 135º at O such that ?AOB = 135º 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
4. AP and BP are the required tangents which make an angle of 45º with each other 
at P. 
Question 4. 
Draw a circle of radius 4.5 cm. Draw two tangents to this circle so that the angle 
between the tangents is 60º. 
 
Solution: 
 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius BC = 4.5 cm 
 
2. Draw arcs making an angle of 180º – 60º = 120º at O such that ?AOB = 120º 
 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
 
4. AP and BP are the required tangents which make an angle of 60º with each other 
at P. 
Question 5. 
Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its 
circumscribed circle. Measure the radius of the circle. 
 
Solution: 
Steps of construction: 
Page 4


Constructions (Circles) 
 
Exercise 19 
Question 1. 
Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the 
circle drawn. Draw two tangents PA and PB to the given circle and measure the length 
of each tangent. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius 3 cm. 
2. From O, take a point P such that OP = 5 cm 
3. Draw a bisector of OP which intersects OP at M. 
4. With centre M, and radius OM, draw a circle which intersects the given circle at A 
and B. 
5. Join AP and BP. 
AP and BP are the required tangents. 
On measuring AP = BP = 4 cm 
Question 2. 
Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre 
of the circle. Draw tangents to the given circle from this exterior point. Measure the 
length of each tangent. 
Solution: 
 
1. Draw a circle of diameter 9 cm, taking O as the centre. 
2. Mark a point P outside the circle, such that PO = 7.5 cm. 
3. Taking OP as the diameter, draw a circle such that it cuts the earlier circle at A and 
B. 
4. Join PA and PB. 
Thus, PA and PB are required tangents. PA = PB = 6 cm 
Question 3. 
Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between 
the tangents is 45º. 
 
Solution: 
 
Steps of Construction: 
 
 
 
1. Draw a circle with centre O and radius BC = 5 cm 
2. Draw arcs making an angle of 180º- 45º = 135º at O such that ?AOB = 135º 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
4. AP and BP are the required tangents which make an angle of 45º with each other 
at P. 
Question 4. 
Draw a circle of radius 4.5 cm. Draw two tangents to this circle so that the angle 
between the tangents is 60º. 
 
Solution: 
 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius BC = 4.5 cm 
 
2. Draw arcs making an angle of 180º – 60º = 120º at O such that ?AOB = 120º 
 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
 
4. AP and BP are the required tangents which make an angle of 60º with each other 
at P. 
Question 5. 
Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its 
circumscribed circle. Measure the radius of the circle. 
 
Solution: 
Steps of construction: 
 
1. Draw a line segment BC = 4.5 cm 
2. With centers B and C, draw two arcs of radius 4.5 cm which intersect each other at 
A. 
3. Join AC and AB. 
4. Draw perpendicular bisectors of AC and BC intersecting each other at O. 
5. With centre O, and radius OA or OB or OC draw a circle which will pass through A, 
B and C. 
This is the required circumcircle of triangle ABC. 
On measuring the radius OA = 2.6 cm 
Question 6. 
Using ruler and compasses only. 
 
(i) Construct triangle ABC, having given BC = 7 cm, AB – AC = 1 cm and ?ABC = 45°. 
(ii) Inscribe a circle in the ?ABC constructed in (i) above. Measure its radius. 
 
Solution: 
 
Steps of Construction: 
 
 
 
i) Construction of triangle: 
Page 5


Constructions (Circles) 
 
Exercise 19 
Question 1. 
Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the 
circle drawn. Draw two tangents PA and PB to the given circle and measure the length 
of each tangent. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius 3 cm. 
2. From O, take a point P such that OP = 5 cm 
3. Draw a bisector of OP which intersects OP at M. 
4. With centre M, and radius OM, draw a circle which intersects the given circle at A 
and B. 
5. Join AP and BP. 
AP and BP are the required tangents. 
On measuring AP = BP = 4 cm 
Question 2. 
Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre 
of the circle. Draw tangents to the given circle from this exterior point. Measure the 
length of each tangent. 
Solution: 
 
1. Draw a circle of diameter 9 cm, taking O as the centre. 
2. Mark a point P outside the circle, such that PO = 7.5 cm. 
3. Taking OP as the diameter, draw a circle such that it cuts the earlier circle at A and 
B. 
4. Join PA and PB. 
Thus, PA and PB are required tangents. PA = PB = 6 cm 
Question 3. 
Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between 
the tangents is 45º. 
 
Solution: 
 
Steps of Construction: 
 
 
 
1. Draw a circle with centre O and radius BC = 5 cm 
2. Draw arcs making an angle of 180º- 45º = 135º at O such that ?AOB = 135º 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
4. AP and BP are the required tangents which make an angle of 45º with each other 
at P. 
Question 4. 
Draw a circle of radius 4.5 cm. Draw two tangents to this circle so that the angle 
between the tangents is 60º. 
 
Solution: 
 
Steps of Construction: 
 
 
1. Draw a circle with centre O and radius BC = 4.5 cm 
 
2. Draw arcs making an angle of 180º – 60º = 120º at O such that ?AOB = 120º 
 
3. AT A and B, draw two rays making an angle of 90º at each point which meet each 
other at point P, outside the circle. 
 
4. AP and BP are the required tangents which make an angle of 60º with each other 
at P. 
Question 5. 
Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its 
circumscribed circle. Measure the radius of the circle. 
 
Solution: 
Steps of construction: 
 
1. Draw a line segment BC = 4.5 cm 
2. With centers B and C, draw two arcs of radius 4.5 cm which intersect each other at 
A. 
3. Join AC and AB. 
4. Draw perpendicular bisectors of AC and BC intersecting each other at O. 
5. With centre O, and radius OA or OB or OC draw a circle which will pass through A, 
B and C. 
This is the required circumcircle of triangle ABC. 
On measuring the radius OA = 2.6 cm 
Question 6. 
Using ruler and compasses only. 
 
(i) Construct triangle ABC, having given BC = 7 cm, AB – AC = 1 cm and ?ABC = 45°. 
(ii) Inscribe a circle in the ?ABC constructed in (i) above. Measure its radius. 
 
Solution: 
 
Steps of Construction: 
 
 
 
i) Construction of triangle: 
? Draw a line segment BC = 7 cm 
? At B, draw a ray BX making an angle of 45º and cut off BE = AB – AC = 1 cm 
? Join EC and draw the perpendicular bisector of EC intersecting BX at A. 
? Join AC. 
?ABC is the required triangle. 
ii) Construction of in circle: 
? Draw angle bisectors of ?ABC and ?ACB intersecting each other at O. 
? From O, draw perpendiculars OL to BC. 
? O as centre and OL as radius draw circle which touches the sides of the ?ABC. 
This is the required in-circle of ?ABC. 
On measuring, radius OL = 1.8 cm 
Question 7. 
Using ruler and compasses only, draw an equilateral triangle of side 5 cm. Draw its 
inscribed circle. Measure the radius of the circle. 
 
Solution: 
Steps of Construction: 
 
 
1. Draw a line segment BC = 5 cm 
2. With centers B and C, draw two arcs of 5 cm radius each which intersect each 
other at A. 
3. Join AB and AC. 
4. Draw angle bisectors of ?B and ?C intersecting each other at O. 
5. From O, draw OL ? BC. 
6. Now with centre O and radius OL, draw a circle which will touch the sides of ?ABC 
On measuring, OL = 1.4 cm