2016 AMC 8

Complete problem set with solutions and individual problem pages

Problem 22 Hard

Rectangle DEFA below is a 3 \times 4 rectangle with DC=CB=BA=1. The area of the "bat wings" (shaded area) is

  • A.

    2

  • B.

    2 \frac12

  • C.

    3

  • D.

    3 \frac12

  • E.

    5

Answer:C

The area of trapezoid CBFE is \frac{1+3}2\cdot 4=8. Next, we find the height of each triangle to calculate their area. The two non-colored isosceles triangles are similar, and are in a 3:1 ratio by AA similarity (alternate interior and vertical angles) so the height of the larger is 3, while the height of the smaller one is 1. Thus, their areas are \frac12 and \frac92. Subtracting these areas from the trapezoid, we get 8-\frac12-\frac92 =\boxed3. Therefore, the answer to this problem is \boxed{\textbf{(C) }3}.

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