Track / Stage 4 / 149 of 340 #409 of 2000

Problem 409

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

3. As shown in Figure 11, a shape is formed by square $ABCD$ and $\triangle BEC$ , where $\angle BEC$ is a right angle. Let the length of $CE$ be $a$ and the length of $BE$ be $b$ . Then the distance from point $A$ to line $CE$ is $\qquad$

Your answer

Official solution

As shown in Figure 11, rotate Rt $\triangle B E C$ 90 degrees clockwise around point $B$ to get Rt $\triangle A B G$ .
Extend $A G$ , intersecting $C E$ at point $F$ . Then $A F \perp C E, A G=C E=a, B G=B E$ .
Therefore, quadrilateral $B G F E$ is a square.
$\begin{array}{l} \Rightarrow G F=B E=b \\ \Rightarrow A G+G F=C E+B E=a+b \end{array}$
$\Rightarrow$ The distance from point $A$ to line $C E$ is $a+b$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.