Radii to Tangents - Problem 1


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.

tangent radius right angle point outside the circle perpendicular