AMC 10 · 2011 · #14
Grade 8 geometry-2dPick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #4 (Introduce a Variable) names the two sides l and w so the diagonal and the area become two equations. Tool #1 (Draw a Diagram) shows why the diagonal makes a right triangle, which turns the diagonal fact into l²+w²=625. Tool #7 (Identify Subproblems) is the key move: the perimeter only needs l+w, not l and w on their own, and (l+w)² = l²+w²+2lw recycles both facts we already have. This dodges the messy work of solving for each side.
Cut the rectangle with its diagonal
The diagonal makes a right triangle whose legs are the sides, so l² + w² = 625.
A rectangle's diagonal always makes a right triangle, so its length ties the two sides together through the Pythagorean theorem.
8.G.B.7Draw A DiagramTurn the area into an equation
Name the sides l and w. Area is length times width, so the area fact reads l · w = 168.
Naming the sides lets each sentence of the problem become one clean equation.
3.MD.C.7Introduce A VariableAim straight at the perimeter
Perimeter needs only l+w, and (l+w)² = l² + w² + 2lw = 625 + 336 = 961.
Squaring the sum of the sides mixes in exactly the two facts you were handed, so no side-by-side solving is needed.
Squaring the sum of the sides mixes in exactly the two facts you were handed.
▸ Why?
Opening the square sends each piece against each piece, giving the two squares and twice the product.
▸ Why?
The diagonal's right angle already ties those two squares to a known length.
Undo the square
A length is positive, so take the positive root of 961: since 31² = 961, l + w = 31 meters.
A square root peels off the square to reveal the sum you were chasing.
8.EE.A.2Introduce A VariableDouble it for the perimeter
The perimeter is 2(l+w), so doubling 31 gives 62 meters — choice (C).
A rectangle's border is just the two sides counted twice, so double their sum.
6.EE.B.7Identify SubproblemsWhen you only need the sum of two sides, square the sum — it reuses the area and diagonal facts you already have, so you never have to find each side.
- Cut the rectangle with its diagonal
- Turn the area into an equation
- Aim straight at the perimeter
- Undo the square
- Double it for the perimeter