AMC 10 · 2024 · #7
Grade 8 geometry-2d
Pick an answer.
The figure is the whole problem: a rectangle cut by the cevians WM and MA into four triangles whose areas must sum to 32. Tool #1 (Draw a Diagram) makes that decomposition visible at a glance. Tool #7 (Identify Subproblems) turns it into two clean conditions — the equal-area condition links XM and ZA, and the right angle at M links them through the Pythagorean theorem. Tool #13 (Convert to Algebra) names XM = b and ZA = a so those two conditions become a 2 × 2 system that pins down a and b. Once a, b are known, the area falls straight out of 32 - (three corner triangles).
Write the three corner areas
Subtract the three corners from the rectangle.
Labeling the picture turns three vague corner triangles into three formulas — the Grade 6 "area by composing/decomposing" idea.
6.G.A.1Draw A DiagramUse the equal areas
It gives b equals two a.
One sentence becomes one equation — Grade 6 "solve real-world problems by writing equations of the form px = q."
6.EE.B.7Convert To AlgebraTurn the right angle into Pythagoras
The three sides give a quadratic.
The right angle at M is a Pythagorean trigger — turn the geometric condition into one numeric equation.
A right angle in the picture turns straight into one numeric equation among the lengths.
▸ Why?
In a right triangle the square on the long side equals the two squares on the legs added together.
▸ Why?
Each corner piece is a triangle whose area is half its base times its height, so the area facts are equations too.
Solve the quadratic
The roots are 1 and 4.
A right angle plus an area condition was always going to collide into a quadratic — Grade 8 linear/quadratic solving handles it.
8.EE.C.7Convert To AlgebraDiscard the outside root
Four lands outside the rectangle.
A root that makes a triangle degenerate is no triangle — kill it. Checking a solution against the picture is Grade 6 "is this value a real solution."
6.EE.B.5Identify SubproblemsSubtract the corners
32 minus the corners is 15.
"Big shape minus the corners" is the cleanest area decomposition — the diagram does all the work.
6.G.A.1Draw A DiagramThis AMC 12 problem only needs Grade 8 Pythagorean theorem plus Grade 6 "rectangle minus the corners" area thinking that you already know!
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