AMC 10 · 2024 · #10
Grade 11 geometry-2dPick an answer.
Neither α nor β has a nice closed form, so computing them separately and subtracting is hopeless. The problem promises a relationship, and the answer choices say what kind: every choice is a whole-number or unit-fraction multiple of α shifted by a constant. Tool #5 (Look for a Pattern) is the move — instead of chasing β, take the multiples of α the choices point at and see what each one looks like. Doubling is the cheapest one to test, and the addition formulas turn 2α into exact fractions with denominator 25, the same denominator the second triangle produces. Tool #1 (Draw a Diagram) then places that doubled angle physically inside the 7-24-25 triangle, which is where the answer actually comes from. Tool #3 (Eliminate Possibilities) keeps the search honest: the constants π/2 and π sitting in the choices are a right angle and a straight angle, so the intended relation is an angle-sum fact, not an arbitrary formula.
Locate both smallest angles
The smallest angle always faces the shortest side.
An angle is pinned down by the ratios of the sides around it, so writing those ratios turns each triangle into a pair of exact numbers.
10.G-SRT.C.6Draw A DiagramRead what the choices promise
Every choice is a piece of π, so expect a special relation.
Constants like π/2 in an answer choice are not decoration — they are a right angle telling you the two angles belong to the same triangle.
11.F-TF.A.1Eliminate PossibilitiesDouble the angle alpha
Doubling spits out 7 and 24 exactly.
Squaring fifths produces twenty-fifths, so doubling an angle of the 3-4-5 triangle can only land on a triangle with hypotenuse 25.
11.F-TF.C.9Look For A PatternFind 2 alpha in the second triangle
2α is the angle facing 24.
Two acute angles with the same sine and cosine are the same angle, so the doubled angle can be redrawn inside the second triangle.
Two acute angles with the same sine and cosine are the same angle.
▸ Why?
Sine and cosine are the two legs of a right triangle with hypotenuse one, so together they pin the shape.
▸ Why?
Between zero and a right angle each pair of values occurs exactly once, so the match is unambiguous.
Acute angles fill the right angle
So β is π/2 minus 2α.
Once both angles live in the same right triangle, nothing is left to compute — the right angle uses up everything they do not.
10.G-CO.C.10Draw A DiagramDoubling the small angle of the 3-4-5 triangle lands you exactly on the big acute angle of the 7-24-25 triangle, and the two acute angles of a right triangle always fill up π/2.
- Locate both smallest angles
- Read what the choices promise
- Double the angle alpha
- Find 2 alpha in the second triangle
- Acute angles fill the right angle