Ex 11.2, 4 (Concept)
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.
Given angle between tangents is 60°
i.e. ∠ QPR = 60°
Since Angle at center is double the angle between tangents
∴ ∠ OQR = 2 × 60° = 120°
So, we need to draw ∠ QOR = 120°
∴ We draw a radius, then second radius at 120° from first.
Also,
Tangent is perpendicular to radius
So, OQ ⊥ QP & OR ⊥ PR
Thus, to make tangents,
we draw perpendicular from point Q and R
So, we draw 90° from point Q and point R
Thus, our figure will look like
Ex 11.2, 4
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.
Steps of construction
Draw a circle of radius 5 cm
Draw horizontal radius OQ
3. Draw angle 120° from point O
Let the ray of angle intersect the circle at point R
Now, draw 90° from point Q
5. Draw 90° from point R
6. Where the two arcs intersect, mark it as point P
∴ PQ and PR are the tangents at an angle of 60°
Justification
We need to prove that PQ and PR are the tangents to the circle at angle of 60° .
Since ∠ PQO = 90°
∴ PQ ⊥ QO
Since tangent is perpendicular to radius,
and QO is the radius
∴ PQ is the tangent to the circle
Similarly,
PR is the tangent to the circle
Now, we prove ∠ P = 60°
In quadrilateral PQOR
Sum of angles = 360°
∠ P + ∠ Q + ∠ R + ∠ QOR = 360°
∠ P + 90° + 90° + 120° = 360°
∠ P + 180° + 120° = 360°
∠ P + 300° = 360°
∠ P = 360° – 300°
∠ P = 60°
So, PQ and PR are tangents at an angle of 60°

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.