2004 AMC 12A Problems/Problem 12

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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 $2\sqrt{2}\ \mathrm{(B)}$ .

2004 AMC 12A (Problems)
Preceded by Problem 11	Followed by Problem 13
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2004 AMC 12A Problems/Problem 12

From AoPSWiki

Problem

Solution

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