1972 AMC 12 Problem 13

Question

Inside square

ABCD

with sides of length

12

inches, segment

AE

is drawn, where

E

is the point on

DC

which is

5

inches from

D.

The perpendicular bisector of

AE

is drawn and intersects

AE,

AD,

and

BC

at points

M,

P,

and

Q,

respectively. The ratio of segment

PM

to

MQ

is:

5{:}12

5{:}13

5{:}19

1{:}4

5{:}21

Draw through

M

a line parallel to

AB

The horizontal distances from

M

to

AD

and

BC

are

\frac{5}{2}

and

\frac{19}{2}

Because

M

is the midpoint of

AE,

its horizontal distance from

AD

is

\frac{5}{2}.

Its distance from

BC

is

12-\frac52=\frac{19}{2}.

The two right triangles cut from line

PQ

by the horizontal through

M

are similar, so their hypotenuses are in the ratio of these horizontal legs:

PM:MQ=\frac52:\frac{19}{2}=5:19.

Therefore, the correct answer is C.

Problem 13 in other years