CBSE Class 10-Mathematics: Chapter – 11 Constructions Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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