1986 AMC 12 Problem 11

Attempt Problem 11 of the 1986 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 1986 AMC 12 solutions, or check the answer key.

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

11.

\triangle ABC,

AB=13,

BC=14

and

CA=15.

Also,

M

is the midpoint of side

AB

and

H

is the foot of the altitude from

A

BC.

The length of

HM

6

6.5

7

7.5

8

Focus on right triangle

AHB

Recall the distance from the midpoint of a right triangle’s hypotenuse to each vertex

Triangle

AHB

is right at

H,

and

M

is the midpoint of its hypotenuse

AB.

The midpoint of a right triangle’s hypotenuse is equidistant from all three vertices, so

HM=\frac{AB}{2}=\frac{13}{2}=6.5.

Thus the correct answer is B.

Problem 11 in other years