In the figure, ABCD is a quadrilateral with right angles at A and C. Points E and F are on AC, and DE and BF are perpendicular to AC. If AE=3, DE=5, and CE=7, then BF=

1990 AMC 12 Problem 20

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20.

In the figure,

ABCD

is a quadrilateral with right angles at

A

and

C.

Points

E

and

F

are on

\overline{AC},

and

\overline{DE}

and

\overline{BF}

are perpendicular to

\overline{AC}.

AE=3,

DE=5,

and

CE=7,

then

BF=

3.6

4

4.2

4.5

5

Place

\overline{AC}

on the

x

-axis with

A=(0,0)

Use dot products for the right angles at

A

and

C

Set

A=(0,0),

E=(3,0),

C=(10,0),

D=(3,-5),

and

B=(f,h),

where

h=BF.

Since

AB\perp AD,

(f,h)\cdot(3,-5)=0,

3f=5h.

Since

BC\perp DC,

(f-10,h)\cdot(-7,-5)=0,

7f+5h=70.

Substitution gives

f=7

and

h=\frac{21}{5}=4.2.

Thus the correct answer is C.