Question

In the obtuse triangle $ABC,$ $AM=MB,$ $MD\perp BC,$ $EC\perp BC.$ If the area of $\triangle ABC$ is $24,$ then the area of $\triangle BED$ is

$9$

$12$

$15$

$18$

not uniquely determined

Answer: B

Concepts:triangle area midpoint area decomposition

Difficulty rating: 1960

Small Hint:

Draw segment $MC$ and compare $\triangle DMC$ with $\triangle DME$

Big Hint:

Those triangles have the same base $MD$ and equal altitudes; also $M$ is the midpoint of $AB$

Solution:

Draw $MC.$ Since $EC\parallel MD,$ points $E$ and $C$ have equal perpendicular distances from $MD,$ so $[DME]=[DMC].$ Therefore $\begin{aligned} [BED]&=[BMD]+[DME]\\ &=[BMD]+[DMC]\\ &=[BMC]. \end{aligned}$ Because $M$ is the midpoint of $AB,$ triangle $BMC$ has half the area of $ABC.$ Thus $[BED]=12.$

Therefore, the correct answer is B.

1984 AMC 12 Problem 26

Answer: B

Difficulty rating: 1960

Small Hint:

Big Hint:

Solution:

Problem 26 in Other Years