2021 AMC 12B Fall Problem 2

Below is the professionally curated solution for Problem 2 of the 2021 AMC 12B Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Fall solutions, or check the answer key.

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Concepts:triangle area area decomposition

Difficulty rating: 810

What is the area of the shaded figure shown below?

$4$

$6$

$8$

$10$

$12$

Solution:

The outer triangle has vertices $(1, 0),$ $(3, 5),$ and $(5, 0),$ giving base $4$ and height $5,$ so its area is $\tfrac12 \cdot 4 \cdot 5 = 10.$

Removed from it is the triangle with vertices $(1, 0),$ $(3, 2),$ and $(5, 0),$ which has base $4$ and height $2,$ so area $\tfrac12 \cdot 4 \cdot 2 = 4.$

The shaded area is $10 - 4 = 6.$

Thus, the correct answer is B.

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