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Radii to Tangents - Problem 3
Given a ray tangent to a point on a circle and the radius of that circle, it is possible to find the length of a segment drawn from the center of the circle to a point outside that circle. Since the ray is tangent to the circle, it is possible to draw a radius that intersects the ray at a right angle. Then, the radius, ray, and segment form a right triangle. Using their side lengths, it is possible to find the length of the segment, which is the hypotenuse of this right triangle.
Recall that by the Pythagorean Theorem, a2 + b2 = c2, where a and b are the lengths of the legs of the right triangle, and c is the length of the hypotenuse. Remember that the hypotenuse is the longest side of a right triangle, and is opposite the right angle. So, simply plug in the lengths of the radius and ray for a and b, then solve for c.
In this problem we’re asked to find this line segment CB. But what are we given?
Well I see that we have a ray CA that’s tangent at point A. I also see that we have a radius BA that’s of length 8. How in the world am I going to find out what this length CB is? Well, one way is to notice that we have a radius to a point of tangency and we know that a radius to a point of tangency always forms a right angle. So let’s go back and label angle A as a right angle.
Now I see that we have a special right triangle, is one of our Pythagorean triples. 8, 15, 17, if CB is 17 then the Pythagorean Theorem is true because 8² plus 15² equals 17².
So to find CB we said, a radius to a point of tangency is a right angle and 8, 15, 17 is one of our Pythagorean triples.