AMC 8 · 2011 · #16

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-trianglespythagorean-theoremisosceles-triangle identify-subproblems ↑ Prerequisites: pythagorean-theoremarea-triangles

📏 Medium solution 💡 3 insights

Problem

Triangle A has sides 25, 25, 30. Triangle B has sides 25, 25, 40. Both are isosceles. Compare the areas A and B.

Pick an answer.

(A)

$A = \dfrac{9}{16}B$

(B)

$A = \dfrac{3}{4}B$

(C)

A = B

(D)

$A = \dfrac{4}{3}B$

(E)

$A = \dfrac{16}{9}B$

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Only side lengths are given, so Tool #1 (Draw a Diagram) is the natural first move: sketch each isosceles triangle and drop the altitude to the unequal side. That altitude bisects the base and creates two right triangles. Tool #7 (Identify Subproblems) then splits the task — compute A from its right-triangle pieces, compute B from its right-triangle pieces, and compare. With a right triangle in each picture, the Pythagorean theorem gives the height and the area follows from $\frac{1}{2}$ × base × height.

1STEP 1

Sketch triangle A (25, 25, 30); the altitude bisects the base into halves of 15, making right triangles with hypotenuse 25 and leg 15.

half-base =

\frac{30}{2}

= 15

2STEP 2

In that right triangle, apply the Pythagorean theorem (leg² + leg² = hyp²) to get the height h_A = 20.

h_A² + 15² = 25² → h_A² = 625 - 225 = 400 → h_A = 20

3STEP 3

Compute area A from base and height: $\frac{1}{2}$ × 30 × 20 = 300.

A =

\frac{1}{2}

× 30 × 20 = 300

4STEP 4

Do the same for triangle B (25, 25, 40): the altitude splits 40 into halves of 20, so the Pythagorean theorem gives height h_B = 15.

h_B² + 20² = 25² → h_B² = 625 - 400 = 225 → h_B = 15

5STEP 5

Area B = $\frac{1}{2}$ × 40 × 15 = 300 — the same as A, so the two areas are equal, choice (C).

B =

\frac{1}{2}

× 40 × 15 = 300 → A = B → (C)

Answer

A = B

The two triangles share the same pair of 25 sides, so the right-triangle pieces both have hypotenuse 25. In triangle A the legs are 15 and 20; in triangle B the legs are 20 and 15 — the same 15-20-25 right triangle (a 3-4-5 scaled by 5) in both pictures. Each full isosceles triangle is built from two of these right triangles, so it makes sense that both areas come out equal to 300. Answer (C) A=B is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 8 Pythagorean-theorem reasoning you already know!

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