1961 AMC 12 Problem 11

Attempt Problem 11 of the 1961 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 1961 AMC 12 solutions, or check the answer key.

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

11.

Two tangents are drawn to a circle from an exterior point

A;

they touch the circle at points

B

and

C,

respectively. A third tangent intersects segment

AB

P

and

AC

R,

and touches the circle at

Q.

AB=20,

then the perimeter of triangle

APR

is:

42

40.5

40

39\dfrac78

not determined by the given information

Tangent segments from the same exterior point have equal lengths

Replace

PQ

PB

and

RQ

RC

in the perimeter

Equal tangent segments give

PQ=PB

and

RQ=RC.

Thus

\begin{aligned} AP+PR+RA &=AP+PQ\\ &\quad+RQ+RA\\ &=AP+PB\\ &\quad+RC+RA\\ &=AB+AC. \end{aligned}

Also

AB=AC=20,

so the perimeter is

40.

Therefore, the correct answer is C.

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