AMC 10 · 2005 · #7

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

pythagorean-theoremangle-sum-trianglearea-rectanglescongruent-triangles spatial-visualizationidentify-subproblemsconvert-to-algebra ↑ Prerequisites: pythagorean-theoremcomplementary-anglesarea-rectangles

📏 Long solution 💡 3 insights 📊 Diagram

Problem

A tilted square sits inside a larger square so that extending each of its sides passes through one corner of the outer square. The outer side is √(50) and one small piece cut off at a corner has length 1. Find the area of the inner square.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Draw a Diagram

The hard part is seeing the picture, not the algebra. Once the four side-lines are drawn out to the four corners, a right triangle appears in each corner of ABCD with a full side of the big square as its hypotenuse. That triangle carries every number in the problem: a leg of 1, a leg built from the unknown side, and a hypotenuse of √(50). So the plan is: name the inner side, prove the four corner triangles are congruent so the short leg is 1 in each, then let the Pythagorean theorem finish it.

1STEP 1

Name the side and trace the lines

Naming the inner side leaves four corner pieces hanging off the vertices.

EF = FG = GH = HE = x, [EFGH] = x²

2STEP 2

Spot the right angle at E

Consecutive inner sides are perpendicular, so each corner holds a right triangle.

∠ BEC = ∠ CFD = 90°, BC = CD = √(50)

3STEP 3

Chase the angles at vertex C

Angle chasing makes two acute angles equal.

∠ BCE + ∠ FCD = 90° = ∠ EBC + ∠ BCE ⟹ ∠ EBC = ∠ FCD

4STEP 4

Match the corner triangles to get CF = 1

The corner triangles are therefore congruent, so the given piece repeats.

△ BEC ≅ △ CFD ⟹ CF = BE = 1 ⟹ CE = x + 1

5STEP 5

Apply the Pythagorean theorem

The Pythagorean theorem then gives the inner side as 6.

1² + (x+1)² = (√(50))² = 50 ⟹ (x+1)² = 49 ⟹ x + 1 = 7 ⟹ x = 6

6STEP 6

Square the side for the area

Squaring gives the area 36, choice (C).

[EFGH] = x² = 6² = 36

Answer

The inner square must fit inside the outer one, and 36 ＜ 50, so the size is plausible. A sharper check uses area bookkeeping: the four corner triangles are congruent right triangles with legs 1 and 7, each of area (1 · 7)/2 = 3.5, and together they fill exactly the region between the squares. Then 36 + 4(3.5) = 36 + 14 = 50, matching the outer area exactly — an independent confirmation that nothing was double-counted. Choice (A) fails this test: an inner side of 5 would make the legs 1 and 6, giving a hypotenuse of √(37), not √(50).

💡Key takeaway

When a tilted square sits inside another, look at the corner triangles — one of them holds every number in the problem.

Name the side and trace the lines
Spot the right angle at E
Chase the angles at vertex C
Match the corner triangles to get CF = 1
Apply the Pythagorean theorem
Square the side for the area