# 2004 AMC 12A Problems/Problem 12

## Problem

Let $A = (0,9)$ and $B = (0,12)$. Points $A'$ and $B'$ are on the line $y = x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C = (2,8)$. What is the length of $\overline{A'B'}$? $\text {(A)} 2 \qquad \text {(B)} 2\sqrt2 \qquad \text {(C)} 3 \qquad \text {(D)} 2 + \sqrt 2\qquad \text {(E)}3\sqrt 2$

## Solution

The equation of $\overline{AA'}$ can be found using points $A, C$ to be $y - 9 = \left(\frac{9-8}{0-2}\right)(x - 0) \Longrightarrow y = -\frac{1}{2}x + 9$. Similarily, $\overline{BB'}$ has the equation $y - 12 = \left(\frac{12-8}{0-2}\right)(x-0) \Longrightarrow y = -2x + 12$. These two equations intersect the line $y=x$ at $(6,6)$ and $(4,4)$. Using the distance formula or $45-45-90$ right triangles, the answer is $2\sqrt{2}\ \mathrm{(B)}$.

## See also

 2004 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

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