NCERT Solutions, Exercise 11.2, Constructions, Class 10 (Mathematics) Class 10 Notes | EduRev

Class 10 : NCERT Solutions, Exercise 11.2, Constructions, Class 10 (Mathematics) Class 10 Notes | EduRev

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Exercise 11.2 
Question 1:  
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, 
construct the pair of tangents to the circle and measure their lengths. Give 
the justification of the construction.  
Answer 1:  
A pair of tangents to the given circle can be constructed as follows.  
Step 1   
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. 
Locate a point P, 10 cm away from O. Join OP.  
Step 2   
Bisect OP. Let M be the mid-point of PO.  
Step 3   
Taking M as centre and MO as radius, draw a circle.  
Step 4   
Let this circle intersect the previous circle at point Q and R.  
Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
 
The lengths of tangents PQ and PR are 8 cm each.  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 6 cm). For this, join OQ and 
OR.  
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Exercise 11.2 
Question 1:  
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, 
construct the pair of tangents to the circle and measure their lengths. Give 
the justification of the construction.  
Answer 1:  
A pair of tangents to the given circle can be constructed as follows.  
Step 1   
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. 
Locate a point P, 10 cm away from O. Join OP.  
Step 2   
Bisect OP. Let M be the mid-point of PO.  
Step 3   
Taking M as centre and MO as radius, draw a circle.  
Step 4   
Let this circle intersect the previous circle at point Q and R.  
Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
 
The lengths of tangents PQ and PR are 8 cm each.  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 6 cm). For this, join OQ and 
OR.  
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?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 2:  
Construct a tangent to a circle of radius 4 cm from a point on the concentric 
circle of radius 6 cm and measure its length. Also verify the measurement 
by actual calculation. Give the justification of the construction.  
Answer 2:  
Tangents on the given circle can be drawn as follows.  
Step 1   
Draw a circle of 4 cm radius with centre as O on the given plane.  
Step 2   
Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this 
circle and join OP.  
Step 3   
Bisect OP. Let M be the mid-point of PO.  
Step 4   
Taking M as its centre and MO as its radius, draw a circle. Let it intersect 
the given circle at the points Q and R.  
Page 3


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Exercise 11.2 
Question 1:  
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, 
construct the pair of tangents to the circle and measure their lengths. Give 
the justification of the construction.  
Answer 1:  
A pair of tangents to the given circle can be constructed as follows.  
Step 1   
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. 
Locate a point P, 10 cm away from O. Join OP.  
Step 2   
Bisect OP. Let M be the mid-point of PO.  
Step 3   
Taking M as centre and MO as radius, draw a circle.  
Step 4   
Let this circle intersect the previous circle at point Q and R.  
Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
 
The lengths of tangents PQ and PR are 8 cm each.  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 6 cm). For this, join OQ and 
OR.  
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?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 2:  
Construct a tangent to a circle of radius 4 cm from a point on the concentric 
circle of radius 6 cm and measure its length. Also verify the measurement 
by actual calculation. Give the justification of the construction.  
Answer 2:  
Tangents on the given circle can be drawn as follows.  
Step 1   
Draw a circle of 4 cm radius with centre as O on the given plane.  
Step 2   
Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this 
circle and join OP.  
Step 3   
Bisect OP. Let M be the mid-point of PO.  
Step 4   
Taking M as its centre and MO as its radius, draw a circle. Let it intersect 
the given circle at the points Q and R.  
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Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
It can be observed that PQ and PR are of length 4.47 cm each.  
In ?PQO,  
Since PQ is a tangent,  
?PQO = 90°  
PO = 6 cm  
QO = 4 cm  
Applying Pythagoras theorem in ?PQO, we obtain  
PQ
2
 + QO
2
 = PQ
2
  
PQ
2
 + (4)
2
 = (6)
2
  
PQ
2 
+ 16 = 36  
PQ
2 
= 36 - 16 
PQ
2 
= 20  
PQ   
PQ = 4.47 cm  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ 
and OR.  
Page 4


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1 
 
Exercise 11.2 
Question 1:  
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, 
construct the pair of tangents to the circle and measure their lengths. Give 
the justification of the construction.  
Answer 1:  
A pair of tangents to the given circle can be constructed as follows.  
Step 1   
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. 
Locate a point P, 10 cm away from O. Join OP.  
Step 2   
Bisect OP. Let M be the mid-point of PO.  
Step 3   
Taking M as centre and MO as radius, draw a circle.  
Step 4   
Let this circle intersect the previous circle at point Q and R.  
Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
 
The lengths of tangents PQ and PR are 8 cm each.  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 6 cm). For this, join OQ and 
OR.  
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2 
 
 
?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 2:  
Construct a tangent to a circle of radius 4 cm from a point on the concentric 
circle of radius 6 cm and measure its length. Also verify the measurement 
by actual calculation. Give the justification of the construction.  
Answer 2:  
Tangents on the given circle can be drawn as follows.  
Step 1   
Draw a circle of 4 cm radius with centre as O on the given plane.  
Step 2   
Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this 
circle and join OP.  
Step 3   
Bisect OP. Let M be the mid-point of PO.  
Step 4   
Taking M as its centre and MO as its radius, draw a circle. Let it intersect 
the given circle at the points Q and R.  
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Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
It can be observed that PQ and PR are of length 4.47 cm each.  
In ?PQO,  
Since PQ is a tangent,  
?PQO = 90°  
PO = 6 cm  
QO = 4 cm  
Applying Pythagoras theorem in ?PQO, we obtain  
PQ
2
 + QO
2
 = PQ
2
  
PQ
2
 + (4)
2
 = (6)
2
  
PQ
2 
+ 16 = 36  
PQ
2 
= 36 - 16 
PQ
2 
= 20  
PQ   
PQ = 4.47 cm  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ 
and OR.  
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(Class – X) 
 
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4 
 
 
?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 3:  
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended 
diameter each at a distance of 7 cm from its centre. Draw tangents to the 
circle from these two points P and Q. Give the justification of the 
construction.  
Answer 3:  
The tangent can be constructed on the given circle as follows.  
Step 1   
Taking any point O on the given plane as centre, draw a circle of 3 cm radius.  
Step 2   
Take one of its diameters, PQ, and extend it on both sides. Locate two points 
on this diameter such that OR = OS = 7 cm  
Step 3   
Bisect OR and OS. Let T and U be the mid-points of OR and OS 
respectively.  
Step 4   
Taking T and U as its centre and with TO and UO as radius, draw two 
circles. These two circles will intersect the circle at point V, W, X, Y 
respectively. Join RV, RW, SX, and SY. These are the required tangents.  
Page 5


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1 
 
Exercise 11.2 
Question 1:  
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, 
construct the pair of tangents to the circle and measure their lengths. Give 
the justification of the construction.  
Answer 1:  
A pair of tangents to the given circle can be constructed as follows.  
Step 1   
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. 
Locate a point P, 10 cm away from O. Join OP.  
Step 2   
Bisect OP. Let M be the mid-point of PO.  
Step 3   
Taking M as centre and MO as radius, draw a circle.  
Step 4   
Let this circle intersect the previous circle at point Q and R.  
Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
 
The lengths of tangents PQ and PR are 8 cm each.  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 6 cm). For this, join OQ and 
OR.  
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2 
 
 
?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 2:  
Construct a tangent to a circle of radius 4 cm from a point on the concentric 
circle of radius 6 cm and measure its length. Also verify the measurement 
by actual calculation. Give the justification of the construction.  
Answer 2:  
Tangents on the given circle can be drawn as follows.  
Step 1   
Draw a circle of 4 cm radius with centre as O on the given plane.  
Step 2   
Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this 
circle and join OP.  
Step 3   
Bisect OP. Let M be the mid-point of PO.  
Step 4   
Taking M as its centre and MO as its radius, draw a circle. Let it intersect 
the given circle at the points Q and R.  
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Step 5   
Join PQ and PR. PQ and PR are the required tangents.  
It can be observed that PQ and PR are of length 4.47 cm each.  
In ?PQO,  
Since PQ is a tangent,  
?PQO = 90°  
PO = 6 cm  
QO = 4 cm  
Applying Pythagoras theorem in ?PQO, we obtain  
PQ
2
 + QO
2
 = PQ
2
  
PQ
2
 + (4)
2
 = (6)
2
  
PQ
2 
+ 16 = 36  
PQ
2 
= 36 - 16 
PQ
2 
= 20  
PQ   
PQ = 4.47 cm  
Justification  
The construction can be justified by proving that PQ and PR are the tangents 
to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ 
and OR.  
(www.tiwariacademy.net) 
(Class – X) 
 
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4 
 
 
?PQO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?PQO = 90°  
? OQ ? PQ  
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. 
Similarly, PR is a tangent of the circle  
 
Question 3:  
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended 
diameter each at a distance of 7 cm from its centre. Draw tangents to the 
circle from these two points P and Q. Give the justification of the 
construction.  
Answer 3:  
The tangent can be constructed on the given circle as follows.  
Step 1   
Taking any point O on the given plane as centre, draw a circle of 3 cm radius.  
Step 2   
Take one of its diameters, PQ, and extend it on both sides. Locate two points 
on this diameter such that OR = OS = 7 cm  
Step 3   
Bisect OR and OS. Let T and U be the mid-points of OR and OS 
respectively.  
Step 4   
Taking T and U as its centre and with TO and UO as radius, draw two 
circles. These two circles will intersect the circle at point V, W, X, Y 
respectively. Join RV, RW, SX, and SY. These are the required tangents.  
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Justification  
The construction can be justified by proving that RV, RW, SY, and SX are 
the tangents to the circle (whose centre is O and radius is 3 cm). For this, 
join OV, OW, OX, and OY.  
  
?RVO is an angle in the semi-circle. We know that angle in a semi-circle is 
a right angle.  
? ?RVO = 90°  
? OV ? RV  
Since OV is the radius of the circle, RV has to be a tangent of the circle. 
Similarly,  
OW, OX, and OY are the tangents of the circle  
 
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