2004 AMC 12A Problem 12

Problem 12 of 25IntermediateGeometry

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'}?

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Solution

Line

AC

passes through

(0, 9)

with slope

\tfrac{8 - 9}{2 - 0} = -\tfrac12,

so its equation is

y = -\tfrac12 x + 9.

Setting

y = x

gives

A' = (6, 6).

Line

BC

passes through

(0, 12)

with slope

-2,

so

y = -2x + 12.

Setting

y = x

gives

B' = (4, 4).

Then

\begin{aligned} A'B' &= \sqrt{(6 - 4)^2 + (6 - 4)^2} \\ &= 2\sqrt{2}. \end{aligned}

Thus, the correct answer is B.

More practice

Concepts: coordinate geometry · distance formula

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.