1967 AMC 12 Problem 5

Attempt Problem 5 of the 1967 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 1967 AMC 12 solutions, or check the answer key.

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

A triangle is circumscribed about a circle of radius

r

inches. If the perimeter of the triangle is

P

inches and the area is

K

square inches, then

\frac{P}{K}

is:

independent of the value of

r

\dfrac{\sqrt2}{r}

\dfrac{2}{\sqrt r}

\dfrac2r

\dfrac r2

Split the triangle into three smaller triangles with altitude

r

Express

K

using the semiperimeter

\frac{P}{2}

The three radii to the points of tangency split the triangle into smaller triangles whose total area is

K=\frac12r(a+b+c)=\frac{Pr}{2}.

Hence

\frac{P}{K}=\frac{2}{r}.

Therefore, the correct answer is D.

Problem 5 in other years