Skip to main content

2009 AMC 12B Problem 13

Problem 13 of 25IntermediateGeometry

Triangle ABCABC has AB=13AB = 13 and AC=15,AC = 15, and the altitude to BCBC has length 12.12. What is the sum of the two possible values of BC?BC?

Answer choices

Show solution

Solution

Let DD be the foot of the altitude from A.A. Then BD=132−122=5BD = \sqrt{13^2 - 12^2} = 5 and DC=152−122=9.DC = \sqrt{15^2 - 12^2} = 9. If DD lies between BB and C,C, then BC=5+9=14BC = 5 + 9 = 14; if the triangle is obtuse, BC=9−5=4.BC = 9 - 5 = 4. The sum is 14+4=18.14 + 4 = 18. Thus, the correct answer is D.

More practice

Concepts: Pythagorean Theorem · altitude · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.