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:

$16n$

$18n$

$21n$

$24n$

$27n$

Answer: D

Concepts:area ratio centroid coordinate geometry median (geometry)

Difficulty rating: 1650

Small Hint:

Use an affine model $A=(0,0),$ $B=(2,0),$ and $C=(0,2)$

Big Hint:

Find $M,O,P,Q$ and compare the two areas

Solution:

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.

1966 AMC 12 Problem 38

Answer: D

Difficulty rating: 1650

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years