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Radii to Tangents - Problem 1

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Given two tangents drawn on a circle that intersect at a given angle, and a radius drawn to each point of tangency, it is possible to find the value of the central angle formed by the two radii. Recall that the radius always intersects the point of tangency at a right angle. So, the two radii form two 90° angles with the drawn tangent lines. From the picture drawn, the shape formed by the two radii and two tangents drawn on the circle is a quadrilateral. Since the sum of the angles in a quadrilateral is 360°, and three angles are known (the two right angles, and the given angle), solve for the unknown angle by subtracting these three angles from 360°.

In this problem we see that we have two tangents that are drawn to the circle. We also see that we have a radius drawn to each point of tangency. So we formed a quadrilateral, but we only know one angle, so how are we going to find X?

To do that we’re going to look and see what do we know about a radius, draw to the point of tangency? Well I see that they’re always going to intersect at a right angle. So I can go back to my problem and I can say that this angle right here, since it’s a radius to a tangent, has to be 90 degrees. This angle right here, since it’s a radius to a tangent, must also be a right angle.

So if the sum of these four angles is 360 degrees, we can write that equation. So we can say 360 is equal to X plus 90 plus 60 plus 90. So now all I need to do is just solve for X. So 360 is equal to 90 plus 90 is 180, 180 plus 60 is 240. So now if I subtract 240 from both sides, subtract 240 degrees, I see that X must be 120 degrees. So I’m going to write that up here, 120 degrees.

The key to this problem is realizing that a radius to a tangent forms a right angle so we actually knew these two angles.

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