Question

To draw a pair of tangents to a circle which are inclined to each other at an angle of $ 50^\circ $, it is required to draw tangents at end points of those two radii of the circle. What should be the angle between these two radii?

• A. $ 150^\circ $
• B. $ 140^\circ $
• C. $ 130^\circ $
• D. $ 120^\circ $

Hint:

The angle between the tangents drawn to a circle from an external point is supplementary to the angle between the radii drawn to the points of tangency. Use the relationship $ \text{Angle between tangents} = 180^\circ - \text{Angle between radii} $ to solve such problems.

Answer 1

The Correct Option is C

Step 1: Understand the Geometry of Tangents and Radii.
When two tangents are drawn to a circle from an external point, the angle between the tangents is related to the angle between the radii drawn to the points of tangency. The relationship is given by: $$ \text{Angle between tangents} = 180^\circ - \text{Angle between radii}. $$ Step 2: Apply the Relationship.
Given that the angle between the tangents is $ 50^\circ $, we can use the formula: $$ \text{Angle between radii} = 180^\circ - \text{Angle between tangents}. $$ Substitute the given angle: $$ \text{Angle between radii} = 180^\circ - 50^\circ = 130^\circ. $$ Step 3: Analyze the Options.
Option (1): $ 150^\circ $ — Incorrect, as this does not match the calculated value.
Option (2): $ 140^\circ $ — Incorrect, as this does not match the calculated value.
Option (3): $ 130^\circ $ — Correct, as it matches the calculated value.
Option (4): $ 120^\circ $ — Incorrect, as this does not match the calculated value. Step 4: Final Answer. $$ (3) \mathbf{130^\circ} $$