2006 AMC 10A Problem 17

Problem 17 of 25IntermediateGeometry

In rectangle

ADEH,

points

B

and

C

trisect

\overline{AD},

and points

G

and

F

trisect

\overline{HE}.

In addition,

AH = AC = 2.

What is the area of quadrilateral

WXYZ

shown in the figure?

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Solution

Set

A = (0, 0),

D = (3, 0),

H = (0, 2),

so

B = (1, 0),

C = (2, 0),

G = (1, 2),

F = (2, 2),

and

E = (3, 2).

The drawn segments meet at

W = (1.5, 1.5),

X = (1, 1),

Y = (1.5, 0.5),

and

Z = (2, 1).

These form a square whose perpendicular diagonals

WY

and

XZ

each have length

1.

Its area is

\frac12 \cdot 1 \cdot 1 = \frac12.

Thus, the correct answer is A.

More practice

Concepts: coordinate geometry · area · square (geometry)

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.