1966 AMC 12 Problem 38

Question

In triangle

ABC

the medians

AM

and

CN

to sides

BC

and

AB,

respectively, intersect in point

O.

P

is the midpoint of side

AC,

and

MP

intersects

CN

in

Q.

If the area of triangle

OMQ

is

n,

then the area of triangle

ABC

is:

16n

18n

21n

24n

27n

Use an affine model

A=(0,0),

B=(2,0),

and

C=(0,2)

Find

M,O,P,Q

and compare the two areas

Area ratios are affine-invariant, so take

A=(0,0),

B=(2,0),

and

C=(0,2).

Then

\begin{gathered} M=(1,1),\\ O=(\frac{2}{3},\frac{2}{3}),\\ P=(0,1). \end{gathered}

Line

MP

is

y=1,

and it meets median

CN

at

Q=(\frac{1}{2},1).

Thus

[OMQ]=\frac{1}{12},

while

[ABC]=2.

Their ratio is

24,

so

[ABC]=24n.

Therefore, the correct answer is D.

Problem 38 in other years