2003 AMC 10A Problem 22

Attempt Problem 22 of the 2003 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10A solutions, or check the answer key.

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22.

In rectangle

ABCD,

we have

AB = 8,

BC = 9,

H

is on

\overline{BC}

with

BH = 6,

E

is on

AD

with

DE = 4,

line

EC

intersects line

AH

at

G,

and

F

is on line

AD

with

\overline{GF} \perp \overline{AF}.

Find the length

\overline{GF}.

16

20

24

28

30

Answer: B

Concepts:coordinate geometry linear equation

Difficulty rating: 1800

Small hint:

Place

D

at the origin with

A

on the positive

x

-axis

Big hint:

Find where line

AH

and line

EC

meet, then

\overline{GF}

is the height of

G

above line

AD

Solution:

Place

D = (0, 0),

A = (9, 0),

B = (9, 8),

C = (0, 8),

H = (3, 8),

and

E = (4, 0).

Line

AH

has equation

y = -\dfrac{4}{3}x + 12,

and line

EC

has equation

y = -2x + 8.

Setting them equal gives

x = -6

and

y = 20,

so

G = (-6, 20).

Since

\overline{GF}

is perpendicular to line

AD

(the

x

-axis), its length is the height

20.

Thus, the correct answer is B.