When semicircles get entangled in squares.

Geometry Level pending

ABCD is a square of side length 2. A semicircle is drawn on side AB. The line CE is tangent to the semicircle and intersects AD at E.

Find the length of EX.

3/4 units 1 unit 1/3 units 1/2 units

1 solution

Chung Kevin
Nov 9, 2016

Let $EX = x$ .

Since tangents to a circle have the same length, thus $EA = EX = x$ and $CX = CB = 2$ .

Applying Pythagorean theorem on triangle $DEC$ ,

$DE^2 + DC^2 = CE^2 \Rightarrow (2-x)^2 + 2^2 = (2+x)^2$

Expanding and simplifying, we obtain $4 = 8x$ or that $x = \frac{1}{2}$ .