AMC 10 · 2009 · #8

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
area-rectanglesratio-proportionsystems-of-equations convert-to-algebraidentify-subproblems ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Four identical rectangles surround a small square and together fill a larger one. The big square's area is four times the small one's. Find the ratio of a rectangle's sides.

Pick an answer.

(A)
3
(B)
$\sqrt {10}$
(C)
$2 + \sqrt2$
(D)
$2\sqrt3$
(E)
4
How to solve
Strategy Introduce a Variable

The figure hides two squares whose sides are built from the rectangle's two side lengths. Name those lengths, read the two square sides off the picture, and the area fact becomes a single equation to solve.

1STEP 1

Read the two squares off the figure

The picture gives both square sides as a sum and a difference.

outer side = ℓ + s, inner side = ℓ - s
2STEP 2

Turn the area fact into an equation

The area fact becomes one equation.

(ℓ + s)² = 4(ℓ - s)²
3STEP 3

Take the square root of both sides

Taking roots makes it linear.

ℓ + s = 2(ℓ - s)
4STEP 4

Solve for the side ratio

Solving gives the ratio 3, choice (E).

ℓ + s = 2ℓ - 2s → 3s = ℓ → ℓ/s = 3
Answer
3
Test with s = 1, ℓ = 3. Outer side = 1 + 3 = 4 and inner side = 3 - 1 = 2, giving areas 16 and 4. Indeed 16 = 4 × 4, so the area condition holds and the ratio 3 is confirmed. It also matches the sample rectangle in the figure.
💡Key takeaway

If one square has 4 times the area of another, its side is twice as long, so turn the area clue into a side clue first.

  • Read the two squares off the figure
  • Turn the area fact into an equation
  • Take the square root of both sides
  • Solve for the side ratio

More like this

Same archetype — closest grade level first.