Problem 14 · 2007 AMC 8
Easy
Geometry & Measurement
isosceles-altitudepythagorean-triple
The base of isosceles ▵ABC is 24 and its area is 60. What is the length of one of the congruent sides?
Show answer
Answer: C — 13.
Show hints
Hint 1 of 2
In an isosceles triangle, the altitude to the base lands dead center, splitting it into two mirror-image right triangles. The area hands you that altitude's length.
Still stuck? Show hint 2 →
Hint 2 of 2
Drop the altitude from the apex: it halves the base and creates a right triangle, turning a side-length question into Pythagoras.
Show solution
Approach: altitude splits it into a right triangle
- Back the height out of the area: 60 = (1/2)(24)(h) ⇒ h = 5.
- That altitude bisects the base, so each right triangle has legs 5 (height) and 12 (half of 24).
- Hypotenuse = the congruent side = √(52 + 122) = √169 = 13.
- Recognize it: 5-12-13 is one of the famous Pythagorean triples — spotting 5 and 12 lets you write 13 instantly, no square root needed.
Mark:
· log in to save