Question

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?

$\dfrac{1}{2}$

$\dfrac{\sqrt{2}}{2}$

$\dfrac{\sqrt{3}}{2}$

$\dfrac{2\sqrt{2}}{3}$

$\dfrac{2\sqrt{3}}{3}$

Answer: A

Concepts:coordinate geometry area square (geometry)

Difficulty rating: 1540

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.

2006 AMC 10A Problem 17

Answer: A

Difficulty rating: 1540

Solution:

Problem 17 in Other Years