# 9.10.6. The Relation between the Tangents of a Circle

Given that , find and .

• A,
• B,
• C,
• D,

### Example

Given that , find and .

### Solution

A line segment passing through the centre of a circle and the point of intersection of two of its tangents bisects the angle between these two tangents. This means that . First, use this fact, along with the fact that , to find the measure of as follows:

Two tangent segments drawn to a circle from a point outside it are equal in length, so is an isosceles triangle. Base angles of an isosceles triangle are congruent, so . Now use this fact, along with the fact that the sum of the measures of the interior angles of a triangle is , to find the measure of .

A tangent segment drawn to a circle and a radius of the circle drawn to the point of tangency are perpendicular, so . Next, use this fact, along with the fact that there are in a quadrilateral, to solve for the measure of .

The measure of an inscribed angle of a circle is half the measure of the central angle subtended by the same arc. Finally, use this fact to find the measure of .

Thus, the measure of is , and the measure of is .

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