AMC 10 · 2005 · #3
Grade 8 geometry-2dPick an answer.
No side length is given as a number, only the ratio of the sides and the diagonal, so Tool #4 (Introduce a Variable) lets us name the width and write both sides — and the area — from that single unknown. Tool #1 (Draw a Diagram) shows why the Pythagorean theorem applies: the diagonal splits the rectangle into a right triangle whose legs are the two sides. Tool #7 (Identify Subproblems) spots the shortcut — the area is 2w², so we only need the value of w², not w itself, and the Pythagorean equation hands us w² directly.
Name the two sides
Naming the short side writes both sides with one unknown.
Naming the width as one variable lets both sides — and the area — grow out of a single unknown.
6.EE.B.6Introduce A VariableDiagonal is the hypotenuse
The diagonal is a hypotenuse, giving one equation.
A rectangle's diagonal always makes a right triangle with the two sides, so the Pythagorean theorem ties the diagonal to them.
8.G.B.7Draw A DiagramSolve for w², not w
Solve for the square of the side, not the side itself.
The area depends on w², so getting w² from the equation is enough — no square roots needed.
The area only ever needs the square of the width, so solving for that square is enough and no root is taken.
▸ Why?
The diagonal, the width and the length form a right triangle, so one equation ties the squared lengths together.
▸ Why?
The same operation on both sides keeps the equation true, so the squared width can be isolated directly.
Compute the area
The area needs exactly that, giving 2/5x², choice (B).
Once w² is known, the area formula 2w² turns straight into an expression in x².
6.EE.A.2Introduce A VariableWhen a shape is described only by its diagonal, name one side as a variable and use the Pythagorean theorem — you can often reach the area from w² without ever finding the side itself.
- Name the two sides
- Diagonal is the hypotenuse
- Solve for w², not w
- Compute the area