CBSE Class 10-Mathematics: Chapter –11 Constructions Part 5
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Question 5:
Draw a pair of tangents to a circle of radius which are inclined to each other at an angle of

A Pair of Tangents
Answer:
To construct: A pair of tangents to a circle of radius which are inclined to each other at an angle of .
Steps of Construction:
(a) Draw a circle of radius with centre O.
(b) Draw an angle of .
(c) At A and B, draw angles which meet at .
Then AC and BC are the required tangents which are inclined to each other at an angle of .
Justification:
and OA is a radius. [By construction]
AC is a tangent to the circle.
and is a radius. [By construction]
BC is a tangent to the circle.Now, in quadrilateral ,
[Angle sum property of a quadrilateral]
Question 6:
Let ABC be a right triangle in which , and is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.

A Right Triangle ABC
Answer:
To construct: A right triangle ABC with , and . BD is the perpendicular from B on AC and the tangents from A to this circle.
Steps of Construction:
(a) Draw a right triangle ABC with , and Also, draw perpendicular BD on AC.
(b) Join AO and bisect it at M (here O is the center of circle through B, C, D).
(c) Taking M as center and MA as radius, draw a circle. Let it intersects the given circle at the points B and E.
(d) Join AB and AE.
Then AB and AE are the required two tangents.
Justification: Join OE.
Then, is an angle in the semicircle.
Since is a radius of the given circle, AE has to be a tangent to the circle. Similarly, AB is also a tangent to the circle.
Question 7:
Prove that the tangents drawn at the ends of a chord of a circle make equal angles with chord.

Prove That the Tangents Drawn at the Ends
Answer:
Let NM be chord of circle with center .
Let tangents at M.N meet at the point .
Since is a tangent
is a tangent
Again in
Thus, tangents make equal angle with the chord