AMC 10 · 2022 · #10
Grade 8 geometry-2d
Pick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) — place the rectangle in coordinates with one cut corner at (0,0) and the other at (l,w). Mark the inner notch-vertices at (1,1) and (l-1, w-1). The picture makes both length facts directly visible: the diagonal of the rectangle is the segment from (0,0) to (l,w), and the segment between the inner notches is from (1,1) to (l-1, w-1) — same direction, just shortened by √(2) at each end. Tool #7 (Identify Subproblems) — turn the two length facts into two equations using the Pythagorean theorem (or distance formula). Tool #13 (Convert to Algebra) — combine the two equations using the identity (l+w)² = l² + w² + 2lw to get lw directly, without ever solving for l and w individually.
Put the card on a grid; the two cut corners sit at (0,0) and (l, w), so their inner notch-vertices are (1,1) and (l-1, w-1).
Sticking the rectangle on a coordinate grid turns each labeled length into a Pythagorean computation.
6.G.A.3Draw A DiagramThe diagonal is the hypotenuse of the right triangle with legs l and w, so Pythagoras with diagonal 8 gives l²+w²=64.
Length-width-diagonal forms a right triangle — the classic Pythagorean setup.
8.G.B.7Identify SubproblemsBetween the inner vertices the changes are l-2 and w-2; length 4√(2) gives (l-2)²+(w-2)²=32 by the distance formula.
Distance between two points in the plane is also a Pythagorean computation on coordinate differences.
8.G.B.8Identify SubproblemsExpand the second equation and substitute l²+w²=64; the squares cancel and it collapses to l+w=10.
Expanding and substituting collapses the two equations into a single linear fact about l+w.
8.EE.C.8Convert To AlgebraFeed l+w=10 and l²+w²=64 into the identity (l+w)²=l²+w²+2lw to read the product lw directly — the area.
The square of a sum identity lets us trade (l+w)² and l²+w² directly for the product lw — the very thing we want (the area).
7.EE.A.1Convert To AlgebraThis AMC 10 problem only needs Grade 8 Pythagorean and algebra you already know — place the rectangle on a grid, write l²+w²=64 and (l-2)²+(w-2)²=32, get l+w=10, and use (l+w)² = l²+w²+2lw to read off area = lw = 18.