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Question

Suppose that we have a right triangle ABC with the right angle at B such that AC = 61 and AB = 5. A circle is drawn with its center on AB such that the circle is tangent to AC and BC. If P is the point where the circle and side AC meet, then what is CP?

Accepted Answer

Since we have a right triangle, we can see that any circle with center on AB is tangent to BC at the right angle itself, or B. Since P is the point at which AC and the circle meet, we see that CP is a tangent from C, as is BC. That means BC = CP. We can easily find BC via the Pythagorean Theorem, as AB^2 + BC^2 = AC^2. Plugging in (5)^2 + BC^2 = (61)^2, we can find BC = CP = 6.