1958 AMC 12 Problem 9

Attempt Problem 9 of the 1958 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 1958 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

A value of

x

satisfying the equation

x^2+b^2=(a-x)^2

is:

\dfrac{b^2+a^2}{2a}

\dfrac{b^2-a^2}{2a}

\dfrac{a^2-b^2}{2a}

\dfrac{a-b}{2}

\dfrac{a^2-b^2}{2}

Expand

(a-x)^2

The

x^2

-terms cancel, leaving a linear equation in

x

Expanding the right side,

x^2+b^2=a^2-2ax+x^2.

Cancelling

x^2

and solving gives

\begin{aligned} 2ax&=a^2-b^2,\\ x&=\frac{a^2-b^2}{2a}. \end{aligned}

Therefore, the correct answer is C.

Problem 9 in other years