Given parallelogram ABCD with E the midpoint of diagonal BD. Point E is connected to a point F in DA so that DF=13DA. What is the ratio of the area of triangle DFE to the area of quadrilateral ABEF?

1964 AMC 12 Problem 22

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22.

Given parallelogram

ABCD

with

E

the midpoint of diagonal

BD.

Point

E

is connected to a point

F

DA

so that

DF=\frac13DA.

What is the ratio of the area of triangle

DFE

to the area of quadrilateral

ABEF?

1:2

1:3

1:5

1:6

1:7

Use vectors

A=0,

B=u,

and

D=v

Then

E=\frac{u+v}{2}

and

F=\frac{2v}{3}

; compute both areas as fractions of the parallelogram

Let the parallelogram’s area be

S=|u\times v|,

with

A=0,

B=u,

and

D=v.

Then

E=\frac{u+v}{2}

and

F=\frac{2v}{3}.

Determinants give

[DFE]=\frac{S}{12}.

For quadrilateral

ABEF,

the two determinant contributions give

[ABEF]=\frac S4+\frac S6=\frac{5S}{12}.

The desired ratio is

1:5.

Therefore, the correct answer is C.