2002 AMC 12A Problem 8

Below is the professionally curated solution for Problem 8 of the 2002 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12A solutions, or check the answer key.

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Concepts:area decomposition congruence (geometry)

Difficulty rating: 1380

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?

$B = W$

$W = R$

$B = R$

$3B = 2R$

$2R = W$

Solution:

Drawing the diagonals shown, the entire flag is tiled by congruent triangles. Counting, there are $24$ triangles in the blue region, $24$ in the white region, and $16$ in the red square.

Since the blue and white regions contain the same number of triangles, $B = W.$

Thus, the correct answer is A.

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