AMC 10 · 2023 · #9

Grade 8 geometry-2d

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

pythagorean-theoremarea-rectanglessimilar-triangles convert-to-algebraidentify-subproblems ↑ Prerequisites: pythagorean-theoremarea-rectangles

📏 Long solution 💡 3 insights 📊 Diagram

Problem

A square of area 3 contains a tilted square of area 2 that touches the outer square once on each side. The tilt cuts four congruent right triangles off the corners. Find the ratio of the shorter leg to the longer leg of one of those triangles.

Pick an answer.

(A)

$\frac15$

(B)

$\frac14$

(C)

$2-\sqrt3$

(D)

$\sqrt3-\sqrt2$

(E)

$\sqrt2-1$

How to solve

Strategy Draw a Diagram

Start with Tool #1 (Draw a Diagram): a quick labeled sketch shows that along any side of the outer square the two legs of one corner triangle sit end-to-end, so leg + leg equals the outer side √(3). The same picture identifies the hypotenuse with a side of the inner square, √(2), so the Pythagorean theorem gives the second equation. Tool #13 (Convert to Algebra) then turns those two pictorial facts into a clean two-equation system in x (short leg) and y (long leg), which solves to a numerical ratio. Finally Tool #3 (Eliminate Possibilities) double-checks the closed-form answer numerically against the five choices.

1STEP 1

Find both side lengths

The areas give each side.

s_out = √(3), s_in = √(2)

2STEP 2

The legs' sum and the hypotenuse

The legs' sum is the outer side.

x + y = √(3), hypotenuse = √(2)

3STEP 3

Use Pythagoras

Pythagoras gives the sum of squares.

x² + y² = (√(2))² = 2

4STEP 4

Find the product

The sum and sum of squares give the product.

(x+y)² = x² + 2xy + y² = 3 → 2xy = 3 - (x²+y²) = 3 - 2 = 1

5STEP 5

Solve for the legs

A quadratic gives both legs.

2t² - 2√(3) t + 1 = 0 → t = (2√(3) ± √(12 - 8))/4 = (2√(3) ± 2)/4 = (√(3) ± 1)/2

6STEP 6

Form the ratio

Divide the shorter by the longer.

x/y = ((√(3)-1)/2)/((√(3)+1)/2) = (√(3)-1)/(√(3)+1)

7STEP 7

Rationalize

Clear the radical from the denominator.

(√(3)-1)/(√(3)+1) · (√(3)-1)/(√(3)-1) = ((√(3)-1)²)/(3-1) = (3 - 2√(3) + 1)/2 = (4 - 2√(3))/2 = 2 - √(3)

8STEP 8

Match the choice

The ratio is two minus root three.

2 - √(3) ≈ 0.268 → (C)

Answer

2-√3

Numeric sanity: x + y = (√(3)-1)/2 + (√(3)+1)/2 = √(3) matches the outer side, and x² + y² = ((√(3)-1)² + (√(3)+1)²)/4 = (4-2√(3)) + (4+2√(3))/4 = 8/4 = 2 matches the squared inner side. Direction check: shorter / longer must be less than 1, and 2 - √(3) ≈ 0.268 ∈ (0,1) — good.

💡Key takeaway

Once you draw the tilted square and label one corner triangle, two facts pop out: the two legs share one outer side (x + y = √(3)), and the hypotenuse is the inner side (x² + y² = 2). From there, Grade 8 algebra of square roots does the rest.