AMC 10 · 2020 · #18

Grade 8 geometry-2d

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

area-trianglessimilar-trianglesisosceles-trianglepythagorean-theorem identify-subproblemsconvert-to-algebracasework ↑ Prerequisites: area-trianglessimilar-trianglespythagorean-theorem

📏 Long solution 💡 3 insights 📊 Diagram

Problem

In a square, two points on the two sides meeting at one vertex sit the same distance from that vertex, and a segment joins them. From two points on the other sides, perpendiculars are dropped onto that segment. The four regions this creates — a triangle, two quadrilaterals, and a pentagon — all have area 1. Find the square of the perpendicular's length.

Pick an answer.

(A)

$\frac{7}{3}$

(B)

$8-4\sqrt2$

(C)

$1+\sqrt2$

(D)

$\frac{7}{4}\sqrt2$

(E)

$2\sqrt2$

How to solve

Strategy Draw a Diagram

Tool #1 (Diagram): place A at the origin and use diagonal AC as the axis of symmetry forced by AE = AH. Tool #7 (Subproblems): the key spot is that FI ∥ GJ (both perpendicular to EH), making FGJI a rectangle — so the pentagon splits cleanly into a right triangle on top and a rectangle below. Tool #9 (Easier Problem): symmetry collapses six unknowns into two (c = CF and p = FI); two equations close the system. Tool #8 (Units/distances) and #13 (Algebra) seal the answer via the diagonal-distance equation and a one-line subtraction.

1STEP 1

Find the side length

An area of four makes the side two.

s = 2, AE = AH = √(2), EH = 2

2STEP 2

Use the symmetry

The two quadrilaterals are congruent.

CF = CG, FI = GJ

3STEP 3

Identify the middle piece

The middle piece is a rectangle.

FGJI is a rectangle with sides FI and IJ

4STEP 4

Write the area equation

One quadrilateral's area gives an equation.

c²/2 + c√(2) FI = 1

5STEP 5

Write the length equation

Following the diagonal gives a second equation.

c + FI √(2) = 4 - √(2)

6STEP 6

Solve the system

Solving gives eight minus four root two.

FI² = 8 - 4√(2) → (B)

Answer

8-4√2

Numerical check: FI² ≈ 2.343, so FI ≈ 1.531. Constraints: FI ＜ distance C-to-EH = 2√(2) - 1 ≈ 1.828 ✓. Recover c = (4 - √(2)) - FI√(2) ≈ 4 - 1.414 - 2.165 = 0.421, inside (0, 2) ✓. Verify pentagon: c²/2 ≈ 0.089, rectangle c√(2) FI ≈ 0.595 · 1.531 ≈ 0.911, sum ≈ 1.000 ✓. Choice (B) confirmed.

💡Key takeaway

This AMC 12 problem only needs Grade 8 geometry — spot that FGJI is a rectangle (because FI ∥ GJ), then the pentagon equation plus the diagonal-distance equation snap together: square the distance, subtract the area, and 2 FI² = 16 - 8√(2) gives FI² = 8 - 4√(2).