AMC 10 · 2023 · #13
Grade 8 geometry-3dPick an answer.
We never need the individual edges — only the symmetric sums s₁=a+b+c and s₂=ab+bc+ca. Tool #7 (Identify Subproblems) chops the job into three rungs: (a) read off s₁ and s₂ from the given totals, (b) plug into the identity (s₁)² = a²+b²+c² + 2s₂ to get a²+b²+c², (c) take the square root via the Pythagorean-theorem extension to find the space diagonal. Tool #13 (Convert to Algebra) handles the identity; Tool #8 (Analyze the Units) confirms the answer is a length (not an area or volume).
Read the sum and the pairwise sum
Two conditions give the sum and pairwise sum.
A rectangular box's surface area and edge total are themselves "sum of edges" and "sum of face products" — exactly the two symmetric sums of a,b,c we need.
6.G.A.4Identify SubproblemsFind the sum of squares
The volume is not needed.
Squaring a+b+c already contains every a²,b²,c² once plus every cross term ab,bc,ca twice — exactly what we need to subtract off.
Squaring the edge total already contains every squared edge once and every cross term twice.
▸ Why?
Expanding a sum multiplied by itself spreads every term across the others.
▸ Why?
The space diagonal squared is exactly the three edge squares added, so that is the piece worth isolating.
Take the diagonal
The square root gives nine quarters.
The space diagonal of a box is the hypotenuse of a right triangle whose legs are a face diagonal and the third edge — applying Pythagoras twice gives d² = a² + b² + c².
8.G.B.7Identify SubproblemsCheck the units
Squares of lengths added, so the units check out.
Tracking units catches errors fast: a diagonal must come out as a length, not a number times an area.
5.MD.A.1Analyze The UnitsYou never need the individual edges a,b,c: the totals give a+b+c=13/4 and 2(ab+bc+ca)=11/2, and the identity (a+b+c)² = a²+b²+c² + 2(ab+bc+ca) delivers a²+b²+c² = 81/16. The space diagonal is √(81/16) = (D) 9/4 — and the volume was a red herring.