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Question

Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects side AD at E. What is the length of CE? FIGURE

Accepted Answer

FIGURE Let the point of tangency be F. By the Two Tangent Theorem BC = FC = 2 and AE = EF = x. Thus DE = 2-x. The Pythagorean Theorem on CDE yields align* DE^2 + CD^2 &= CE^2 (2-x)^2 + 2^2 &= (2+x)^2 x^2 - 4x + 8 &= x^2 + 4x + 4 x &= 12align* Hence CE = FC + x = 52 52.