2009 AMC 12B Problem 13 — Solution | LIVE by Po-Shen Loh

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2009 AMC 12B Problem 13

Attempt Problem 13 of the 2009 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

Triangle $ABC$ has $AB = 13$ and $AC = 15,$ and the altitude to $BC$ has length $12.$ What is the sum of the two possible values of $BC?$

$15$

$16$

$17$

$18$

$19$

Answer: D

Concepts:Pythagorean Theorem altitude casework

Difficulty rating: 1560

Solution:

Let $D$ be the foot of the altitude from $A.$ Then $BD = \sqrt{13^2 - 12^2} = 5$ and $DC = \sqrt{15^2 - 12^2} = 9.$

If $D$ lies between $B$ and $C,$ then $BC = 5 + 9 = 14$ ; if the triangle is obtuse, $BC = 9 - 5 = 4.$ The sum is $14 + 4 = 18.$

Thus, the correct answer is D.

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