1976 AMC 12 Problem 9

Attempt Problem 9 of the 1976 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 1976 AMC 12 solutions, or check the answer key.

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

In triangle

ABC,

D

is the midpoint of

AB;

E

is the midpoint of

DB;

and

F

is the midpoint of

BC.

If the area of

\triangle ABC

96,

then the area of

\triangle AEF

16

24

32

36

48

Compare the base

AE

with

AB

and the height from

F

with the height from

C

The two ratios are

\frac34

and

\frac12

Since

D

is the midpoint of

AB

and

E

is the midpoint of

DB,

AE=\frac34AB.

Since

F

is the midpoint of

BC,

its distance from line

AB

is half that of

C.

Therefore

\begin{aligned} [AEF]&=\frac34\cdot\frac12[ABC]\\ &=\frac38(96)=36. \end{aligned}

Therefore, the correct answer is D.

Problem 9 in other years