Multiple choice

To draw a pair of tangents to a circle which are inclined to each other at an angle $\displaystyle { 35 }^{ \circ }$, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is

  1. $\displaystyle { 105 }^{ \circ }$
  2. $\displaystyle { 70 }^{ \circ }$
  3. $\displaystyle { 145 }^{ \circ }$
  4. $\displaystyle { 150 }^{ \circ }$
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C Correct answer
Explanation

The angle between two tangents to a circle and the angle between the radii at the points of contact are supplementary. If the tangents are inclined at 35 degrees, the angle between the radii is 180 - 35 = 145 degrees.

AI explanation

When two tangents intersect at an external point, the line from the center to that point bisects the angle between them, creating an angle of 17.5 degrees with each tangent. Since a radius is perpendicular to a tangent at the point of contact, a right triangle is formed where the angle between the radius and the central line is 90 minus 17.5 equals 72.5 degrees. The angle between the two radii is twice this value because the center line acts as a common side for both symmetrical right triangles. Multiplying 72.5 by 2 gives the required central angle of 145 degrees.

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