2004 AMC 12A Problem 18

Problem 18 of 25IntermediateGeometry

Square

ABCD

has side length

2.

A semicircle with diameter

\overline{AB}

is constructed inside the square, and the tangent to the semicircle from

C

intersects side

\overline{AD}

E.

What is the length of

\overline{CE}?

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Solution

Let

F

be the point where

CE

touches the semicircle. Since

CB

and

CF

are both tangents from

C,

we have

CF = CB = 2.

Similarly, with

x = AE,

the tangents from

E

give

EF = EA = x.

Thus

CE = CF + FE = 2 + x.

In right triangle

CDE,

where

CD = 2

and

DE = 2 - x,

(2 - x)^2 + 2^2 = (2 + x)^2.

Expanding gives

8x = 4,

x = \tfrac12

and

CE = 2 + \tfrac12 = \tfrac52.

Thus, the correct answer is D.