AMC 10 · 2024 · #22
Grade 10 geometry-2dnumber-theoryPick an answer.
The condition lives on the angles, but the quantity to minimise lives on the sides, and integers can only be hunted on the side of the fence. So tool #13 (Convert to Algebra) sets the goal: replace ∠ B = 2∠ A by an exact equation in a, b, c. Tool #1 (Draw a Diagram) supplies the mechanism, because the word "twice" is an instruction to cut ∠ B in half and create a second copy of ∠ A inside the figure. Tool #15 (Organize Information in More Ways) reads the resulting picture as two nested similar triangles, which prices every new segment in terms of a, b, c. Tool #11 (Work Backwards) then runs the same diagram in reverse, so the side equation is proved equivalent to the angle condition rather than merely implied by it. Finally tool #14 (Extreme Principle) traps the perimeter between 2b and 3b, which caps b at a single digit, and tool #2 (Make a Systematic List) sweeps the handful of survivors.
Name each side after its angle
Label the sides a, b, c.
You cannot search for integers among angles, so the angle fact has to be rewritten as a fact about lengths.
9.A-CED.A.2Introduce A VariableCut angle B into two copies of A
Bisecting creates an isosceles triangle.
Halving the doubled angle manufactures a second copy of ∠ A inside the figure, and two equal angles always leave an isosceles triangle behind.
Halving the doubled angle manufactures a second copy of the small angle, and two equal angles leave an isosceles triangle.
▸ Why?
An angle can be split into pieces that add back to it, so the doubled angle really does contain two copies.
▸ Why?
Two equal angles in a triangle face two equal sides, so the split hands over a length equality.
Spot the shrunken copy
The small one is similar to the whole.
The small triangle is the whole triangle scaled by a/b, so every fresh length is an old length times that one factor.
10.G-SRT.A.2Organize Information In More WaysMeasure AD two ways
Equating gives b squared equals a times a plus c.
Computing one segment by two different routes is the standard way a picture becomes an equation.
9.A-CED.A.1Convert To AlgebraCheck the trade runs both ways
The converse holds, so the conditions are equivalent.
Reading the same diagram backwards shows the side equation is not a consequence of the angle condition but a restatement of it.
10.G-SRT.B.5Work BackwardsRead off which integers qualify
b must sit between a and twice a.
Once c is written from a and b, all three triangle inequalities collapse into the single sandwich a ＜ b ＜ 2a.
9.A-CED.A.3Convert To AlgebraTrap the perimeter
The perimeter lies between two b and three b.
Squeezing the perimeter against the middle side turns an unlimited search into a search over a handful of values of b.
9.A-REI.B.3Extreme PrincipleCheck every allowed value and finish
The 4-6-5 triangle gives perimeter 15.
With b capped in the single digits, the last question is simply which number just below b divides b².
6.NS.B.4Make A Systematic ListWhen one angle is twice another, cut the big one in half: the extra copy of the small angle it creates turns the angle fact into a plain equation between the side lengths.
- Name each side after its angle
- Cut angle B into two copies of A
- Spot the shrunken copy of the triangle
- Measure AD two ways
- Check that the trade runs both ways
- Read off exactly which integers qualify
- Trap the perimeter between 2b and 3b
- Check every allowed b and finish