AMC 10 · 2005 · #8

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
pythagorean-theoremarea-rectangles identify-subproblemsconvert-to-algebra ↑ Prerequisites: pythagorean-theoremperfect-squares
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A large square ABCD has side length √(50). Four congruent right triangles are folded inside, leaving a smaller tilted square EFGH in the middle. Point E sits on segment BH with BE = 1. Find the area of the inner square EFGH.

Pick an answer.

(A)
25
(B)
32
(C)
36
(D)
40
(E)
42

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The inner square's side is not handed to us directly, so break the picture into pieces you can measure. Tool #1 shows the outer square is really the inner square plus four congruent right triangles, and each outer side √(50) is a triangle's hypotenuse with short leg BE=1. Tool #7 then splits the job into two easy subproblems: first find the triangle's long leg with the Pythagorean theorem, then notice the long leg is exactly the inner-square side plus one more short leg. Tool #4 lets you call the long leg a name and write 1² + b² = 50.

1STEP 1

Decompose the figure

The four corner triangles are congruent, so in △ BEC the hypotenuse is BC = √(50) and the short leg is BE = 1.

BC = √(50), BE = 1, ∠ BEC = 90°
2STEP 2

Find the long leg

Call the long leg b = EC. Pythagoras gives 1² + b² = 50, so b² = 49 and the long leg is EC = 7.

1² + b² = 50 → b² = 49 → b = √(49) = 7
3STEP 3

Get the inner-square side

Vertex F splits EC into EF plus FC, and FC = 1 is the neighbor triangle's short leg, so the inner side is EF = 6.

EF = EC - FC = 7 - 1 = 6
4STEP 4

Compute the area

Square the inner side: the area of EFGH is 6² = 36, choice (C).

[EFGH] = 6² = 36 → (C)
Answer
36
Add the pieces back up: four right triangles of legs 1 and 7 have total area 4 · 1/2 · 1 · 7 = 14, and the inner square has area 36, giving 14 + 36 = 50 — exactly the outer square's area (√(50))² = 50. Everything fits, so 36 (C) is right. The choice 25 (A) would be the inner side squared if you forgot to subtract the short leg (7-1 mistaken for 5), and 42 (E) is a distractor with no clean derivation.
💡Key takeaway

Cut the big square into an inner square plus four matching right triangles, use the Pythagorean theorem to get the long leg, and the inner side is just the long leg minus the short one.

  • Decompose the figure
  • Find the long leg
  • Get the inner-square side
  • Compute the area

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