AMC 10 · 2015 · #19
Grade 8 geometry-2dPick an answer.
Two right angles let me set the shape on a grid and name the unknown sides with letters. Once a letter stands for BC and a letter stands for CD, the equal sides CD = AD turn into one Pythagorean equation. Solving that equation reveals which side lengths are allowed, and a boundary check finds how many fit under the perimeter limit.
Set the shape on a grid
A grid makes both right angles explicit.
Two right angles act like the corner of graph paper, so the corners get clean coordinates.
6.G.A.3Draw A DiagramName the unknown sides
Two letters name the unknown sides.
Giving the unknown lengths letters lets the equal sides talk to each other in one equation.
6.EE.B.6Introduce A VariableApply the Pythagorean theorem
One right triangle links them.
The equal sides CD and AD are the two equal sides of a right triangle's setup, so Pythagoras pins them together.
8.G.B.7Introduce A VariableSimplify to a square condition
Whole numbers force a perfect square.
A square root only lands on a whole number when what is inside is a perfect square.
A square root lands on a whole number only when what is inside is a perfect square.
▸ Why?
A perfect square needs every prime in its recipe an even number of times, and that is a checkable condition.
▸ Why?
The quantity under the root came from the two legs of a right triangle, so it is fixed by the side lengths.
Write the perimeter as a formula
The perimeter becomes one clean formula.
Each allowed value of k builds one genuine shape, so counting k counts perimeters.
6.EE.B.6Introduce A VariableFind the boundary under 2015
The bound is crossed at 31, choice (B).
Push k up until the perimeter just crosses 2015; the last k that fits sets the count.
7.EE.B.4Extreme PrincipleName the mystery sides with letters, let the right angles give you a Pythagorean equation, and the only shapes that work line up one per whole number k.
- Set the shape on a grid
- Name the unknown sides
- Apply the Pythagorean theorem
- Simplify to a square condition
- Write the perimeter as a formula
- Find the boundary under 2015