AMC 10 · 2024 · #21
Grade 11 geometry-2dnumber-theoryPick an answer.
An equation about angles is useless on its own — α + β + γ = 90° cannot be solved by staring at it, because the two known angles are ugly numbers like 36.87° that no triple hands over exactly. The move is to stop working with the angles and work with their tangents instead (Convert to Algebra): in a right triangle the tangent of the smallest angle is just short leg over long leg, a clean fraction the triple gives for free. The tangent addition formula turns the angle sum into arithmetic on fractions, and the 90° total is the lucky part — a complementary angle flips tangent into its reciprocal (Work Backwards from the total). That pins the third triangle's leg ratio exactly. Naming the scale factor (Introduce a Variable) and demanding primitivity then forces the actual side lengths, after which the hypotenuse and perimeter are small separate computations (Identify Subproblems), with the answer list available as a final cross-check (Eliminate Possibilities).
Turn each triangle into a tangent
The tangents are three quarters and five twelfths.
The angles themselves are messy decimals, but their tangents are exactly the fractions printed in the triple.
10.G-SRT.C.6Draw A DiagramRead the sum as a complement
The third is the complement of the first two.
Two known angles eat part of the right angle; the third triangle is defined by exactly the leftover.
4.MD.C.7Work BackwardsAdd the two known angles
The formula gives 56 over 33.
Adding two angles is hard, but adding their tangents through one formula is just fraction arithmetic.
11.F-TF.A.2Convert To AlgebraFlip for the complementary angle
Flipping gives 33 over 56.
Standing at the other acute corner of a right triangle swaps which leg is opposite and which is adjacent, so the tangent turns upside down.
Standing at the other acute corner of a right triangle swaps which leg is opposite, so the tangent turns upside down.
▸ Why?
A tangent is the far leg over the near one, and the two acute corners trade those two legs.
▸ Why?
Swapping the top and bottom of a fraction is exactly taking its reciprocal.
Primitive forces the scale to be one
They are coprime, so the scale is 1.
The ratio already sits in lowest terms, so the smallest whole-number version of this shape is the only primitive one.
6.NS.B.4Introduce A VariableFind the hypotenuse
The hypotenuse is 65.
The ratio gave the shape and primitivity gave the size, so the third side is now a single square-root away.
8.G.B.7Identify SubproblemsAdd the sides
33 plus 56 plus 65 is 154.
Every choice looks like a legal triple, so the angle condition is the only thing doing real work.
4.NBT.B.4Eliminate PossibilitiesAngles that add to a right angle are hard to handle directly, but their tangents are exact fractions the triples hand you — add the tangents with one formula, flip the result because the leftover angle is complementary, and the ratio you get is the third triangle itself.
- Turn each triangle into a tangent
- Read the sum as a complement
- Add the two known angles
- Flip for the complementary angle
- Primitive forces the scale to be one
- Find the hypotenuse
- Add the sides