Let A = (0, 9) and B = (0, 12). Points A' and B' are on the line y = x, and AA' and BB' intersect at C = (2, 8). What is the length of A'B'?

2004 AMC 12A Problem 12

Attempt Problem 12 of the 2004 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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12.

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'}?

2

2\sqrt{2}

3

2 + \sqrt{2}

3\sqrt{2}

Find the equations of lines

AC

and

BC,

then intersect each with

y = x

A'

and

B'

both lie on

y = x,

so each has equal coordinates

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,

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.