1957 AMC 12 Problem 37

Attempt Problem 37 of the 1957 AMC 12 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 1957 AMC 12 solutions, or check the answer key.

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37.

In right triangle

ABC,

BC=5,

AC=12,

and

AM=x;

MN\perp AC,

NP\perp BC;

N

is on

\overline{AB}.

If

y=MN+NP,

one-half the perimeter of rectangle

MCPN,

then:

y=\dfrac12(5+12)

y=\dfrac{5x}{12}+\dfrac{12}{5}

y=\dfrac{144-7x}{12}

y=12

y=\dfrac{5x}{12}+6

Answer: C

Concepts:similarity rectangle linear equation

Difficulty rating: 1590

Small hint:

Triangles

AMN

and

ACB

are similar

Big hint:

Use

\frac{MN}{x}=\frac{5}{12}

and

NP=MC=12-x

Solution:

By similarity,

MN=\frac5{12}AM=\frac{5x}{12}.

Also

NP=MC=AC-AM

=12-x.

Hence

\begin{aligned} y=MN+NP &=\frac{5x}{12}+12-x\\ &=\frac{144-7x}{12}. \end{aligned}

Therefore, the correct answer is C.

Problem 37 in other years