AMC 8 · 2007 · #14

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

area-trianglespythagorean-theoremisosceles-triangle identify-subproblems ↑ Prerequisites: area-trianglespythagorean-theorem

📏 Short solution 💡 2 insights

Problem

Isosceles △ ABC has base 24 and area 60. Find the length of one of the two congruent sides.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) is the natural opening: sketch the isosceles triangle and drop the altitude from the apex to the base. Because the triangle is isosceles, that altitude bisects the base, splitting the figure into two congruent right triangles. The picture turns the problem into a Pythagorean Theorem exercise where the congruent side is the hypotenuse. Tool #5 (Look for a Pattern) is the finisher: once the two legs are 5 and 12, recognizing the classic 5-12-13 Pythagorean triple gives the hypotenuse instantly — no square root needed.

1STEP 1

Drop the altitude from apex A to base BC; in an isosceles triangle it bisects the base, so BM = MC = 12 and a right angle appears at M.

BM = MC =

\frac{24}{2}

= 12, ∠ AMB = 90°

2STEP 2

Read the area formula backwards: 60 = $\frac{1}{2}$ × 24 × AM forces the altitude AM = 5.

60 =

\frac{1}{2}

× 24 × AM → 60 = 12 · AM → AM = 5

3STEP 3

△ AMB has legs AM = 5 and BM = 12, so hypotenuse AB is the 5-12-13 triple's 13 — no square root needed.

AB² = 5² + 12² = 25 + 144 = 169 → AB = 13 → (C)

Answer

Check the triangle inequality: sides 13, 13, 24 satisfy 13 + 13 = 26 ＞ 24, so the triangle exists (barely — it is long and flat). Recompute the area to confirm: with base 24 and altitude 5, area =

\frac{1}{2}

· 24 · 5 = 60, which matches. Among the choices, 5 is the altitude (a distractor), 8 is half the perimeter minus the base (a distractor), 14 and 18 would force altitudes of √(14² - 12²) = √(52) and √(18² - 12²) = √(180), neither matching 5. Only 13 fits.

💡Key takeaway

Drop the altitude in an isosceles triangle and it bisects the base — the picture hands you a right triangle, and the 5-12-13 Pythagorean triple finishes the problem.

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