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.

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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

The foot of the altitude splits

BC

; use the Pythagorean theorem in the two right triangles

The foot distances are

\sqrt{13^2 - 12^2} = 5

and

\sqrt{15^2 - 12^2} = 9

;

BC

is their sum or difference

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.

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.

Problem 13 in other years