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Radii to Tangents - Problem 3
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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**, a^{2} + b^{2} = c^{2}, 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.

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