2024 AMC 10B Problem 10

Problem 10 of 25EasierGeometry

Quadrilateral

ABCD

is a parallelogram, and

E

is the midpoint of the side

\overline{AD}.

Let

F

be the intersection of lines

EB

and

AC.

What is the ratio of the area of quadrilateral

CDEF

to the area of triangle

CFB?

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Solution

Area ratios don’t change under an affine map, so drop in convenient coordinates:

A = (0,0),

B = (1,0),

C = (1,1),

D = (0,1),

which makes

E = (0, \tfrac12).

Line

AC

y = x,

and line

EB

runs from

(0, \tfrac12)

(1, 0);

they cross at

F = (\tfrac13, \tfrac13).

The shoelace formula gives quadrilateral

CDEF

area

\tfrac{5}{12}

and triangle

CFB

area

\tfrac13.

So the ratio is

\tfrac{5}{12} : \tfrac13 = 5 : 4.

Therefore, the answer is A.