AMC 10 · 2025 · #21
Grade 11 geometry-2dPick an answer.
Two sides are pinned at 8 and 9, so the only freedom left in each triangle is the angle squeezed between them. Naming that angle (Introduce a Variable) turns both the area and the third side into expressions in one unknown, and a quick sketch of the two triangles (Draw a Diagram) shows why one must be squat and the other tall. Equal area then forces a relation between the two angles, the Law of Cosines converts that relation into a single equation with no angle left in it, and what remains is a small integer hunt: bound the third side with the triangle inequality (Eliminate Possibilities) and scan the surviving candidates (Make a Systematic List) to prove exactly one pair works, then push that pair back through the Law of Cosines (Guess and Check) to confirm it really builds a triangle.
Name the angle between 8 and 9
The included angle decides everything.
With two sides locked, a triangle is just a hinge — the angle at the hinge is the only thing that can change.
7.G.A.2Draw A DiagramEqual area forces equal sines
Equal areas force equal sines.
The height of the triangle is the side of length 9 tilted by the hinge angle, so area depends on the angle only through its sine.
10.G-SRT.C.8Introduce A VariableNon-congruent makes them supplementary
Not congruent means they are supplementary.
Sine cannot tell an angle apart from its supplement, so equal area leaves exactly one alternative shape.
A sine cannot tell an angle apart from its supplement, so equal area leaves exactly one alternative shape.
▸ Why?
An angle and its supplement together fill a straight angle, and both open the same height.
▸ Why?
The cosine does tell them apart, which is exactly why the two third sides come out different.
Law of Cosines on both triangles
The cosine law gives two formulas differing only in sign.
Cosine, unlike sine, does distinguish an angle from its supplement — that is exactly what makes the two third sides differ.
11.G-SRT.D.11Introduce A VariableAdd the equations to kill the angle
Adding kills the angle: the squares sum to 290.
The unknown angle appears in the two equations with opposite signs, so adding them makes it vanish for free.
9.A-REI.C.6Introduce A VariableBound each third side
The triangle inequality traps each side between 2 and 16.
A side that equals the sum or the difference of the other two flattens the triangle, so those endpoints are never allowed.
7.G.A.2Eliminate PossibilitiesScan for two squares summing to 290
The only squares summing to 290 are 11 and 13.
Remainders mod 4 throw away half the candidates before any arithmetic is done.
8.EE.A.2Make A Systematic ListConfirm the pair and add
Verified and added, that is 24.
A candidate pair is only real if the cosine it demands is a number a genuine angle can have.
11.G-SRT.D.11Guess And CheckWhen two sides are fixed, the angle between them is the only dial — equal area means the two triangles use an angle and its supplement, and since cosine flips sign between them, the two third sides squared always add up to the same fixed number.
- Name the angle between 8 and 9
- Equal area forces equal sines
- Non-congruent makes them supplementary
- Law of Cosines on both triangles
- Add the equations to kill the angle
- Bound each third side
- Scan for two squares summing to 290
- Confirm the pair and add