1999 AMC 12 Problem 22

Below is the professionally curated solution for Problem 22 of the 1999 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.

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Concepts:absolute value coordinate geometry midpoint

Difficulty rating: 1740

22.

The graphs of $y = -|x - a| + b$ and $y = |x - c| + d$ intersect at points $(2, 5)$ and $(8, 3).$ Find $a + c.$

$7$

$8$

$10$

$13$

$18$

Solution:

The first graph is an inverted right angle with vertex $(a, b),$ and the second is an upright right angle with vertex $(c, d).$ Because each consists of two lines of slope $\pm 1,$ the four points $(a, b), (2, 5), (c, d), (8, 3)$ are the vertices of a rectangle in order.

The diagonals of a rectangle share a midpoint, so $\dfrac{a + c}{2} = \dfrac{2 + 8}{2} = 5,$ giving $a + c = 10.$

Thus, the correct answer is C.

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