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Problem 357

# The angle formed by two tangents drawn to a circle from the same external point measures $$80^{\circ}$$. Find the measure of the minor intercepted arc.

Expert verified
The measure of the minor intercepted arc of the tangents drawn to a circle from the same external point is $$160^{\circ}$$.
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## Step 1: Draw a diagram

Draw a circle with center O and two tangents, PA and PB, from an external point P. Label the points where the tangents touch the circle as A and B. The circle, the tangents, and the intercepted arc AB are shown in the diagram below. ![Circle with tangents PA and PB](https://mathworld.wolfram.com/images/eps-gif/TangentExternalAngleTheorem_1000.gif)

## Step 2: Use the tangent property to find the measures of angles APB and AOB

One of the properties of tangents states that the tangent at a point on a circle is perpendicular to the radius through that point. Therefore, we have: Angle OAP = Angle OBP = $$90^{\circ}$$ We are given the measure of angle APB as $$80^{\circ}$$. Since the sum of angles in a triangle is $$180^{\circ}$$, we can find the measure of angle AOB as follows: Angle AOB = Angle OAP + Angle APB + Angle OBP Angle AOB = $$90^{\circ} + 80^{\circ} + 90^{\circ}$$ Angle AOB = $$260^{\circ}$$

## Step 3: Use the property of inscribed angles to find the measure of the minor intercepted arc

We know that the measure of an inscribed angle (an angle with its vertex on the circle) is half the measure of the intercepted arc. In this case, we have: Insribed Angle = Angle APB $$= 80^{\circ}$$ Using the property of inscribed angles, we can find the measure of the minor intercepted arc AB as follows: Measure of intercepted arc AB = 2 x Insribed Angle Measure of intercepted arc AB = 2 x $$80^{\circ}$$ Measure of intercepted arc AB = $$160^{\circ}$$

## Step 4: State the answer

The measure of the minor intercepted arc of the tangents drawn to a circle from the same external point is $$160^{\circ}$$.

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