Let line AC be perpendicular to line CE. Connect A to the midpoint D of CE, and connect E to the midpoint B of AC. If AD and EB intersect in point F, and BC=CD=15 inches, then the area of triangle DFE, in square inches, is:

1965 AMC 12 Problem 16

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16.

Let line

AC

be perpendicular to line

CE.

Connect

A

to the midpoint

D

CE,

and connect

E

to the midpoint

B

AC.

AD

and

EB

intersect in point

F,

and

BC=CD=15

inches, then the area of triangle

DFE,

in square inches, is:

50

50\sqrt2

75

\dfrac{15}{2}\sqrt{105}

100

Since

B,D

are midpoints and

BC=CD=15,

both perpendicular legs have length

30

Use coordinates

C=(0,0), A=(0,30)

and

E=(30,0)

; the two medians meet at the centroid

Put

C=(0,0),\ A=(0,30),

and

E=(30,0).

Then

B=(0,15)

and

D=(15,0).

Lines

AD

and

EB

are medians of triangle

ACE,

so they meet at the centroid

F=(10,10).

Segment

DE

has length

15

and

F

10

units above its line. Therefore

[DFE]=\frac12(15)(10)=75.

Thus, the correct answer is C.