AMC 10 · 2009 · #14
Grade 8 geometry-2d
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Read the two squares off the figure
One edge of the big square is a full long side plus a short side; the leftover gap in the middle is a long side minus a short side.
Walking along the big square's edge, you cross one long side and one short side end to end.
6.G.A.1Draw A DiagramTurn the area fact into an equation
Call the shorter side s and the longer side ℓ. Area is side squared, so (ℓ + s)² must be 4 times (ℓ - s)².
Both regions are squares, so 'area = side squared' converts the picture into one clean equation.
6.EE.B.6Introduce A VariableTake the square root of both sides
Both sides are squares of positive lengths and 4 = 2², so the positive square root clears the exponents.
Areas in ratio 4 : 1 mean side lengths in ratio 2 : 1.
Areas in the ratio four to one mean side lengths in the ratio two to one.
▸ Why?
Area grows as the square of the length, so a ratio of areas is the square of the ratio of sides.
▸ Why?
Taking the root of both sides of a true equation keeps it true, and lengths are positive.
Solve for the side ratio
Expanding gives ℓ + s = 2ℓ - 2s, so ℓ = 3s and the longer side is triple the shorter — choice (A).
Once the exponents are gone, it is a one-line linear equation in the two lengths.
8.EE.C.7Convert To AlgebraIf 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