AMC 10 · 2022 · #19
Grade 11 geometry-2dPick an answer.
The problem hands over no numbers at all, only a shape condition, so the first move is to supply the missing number yourself. Call the side of the equilateral triangle s. That one name is enough, because every other length in the picture is chained to it: E is a midpoint, so AC = 2s immediately, and the centroid's 2:1 rule converts AG and GE into the full medians and their remaining pieces. At the end s cancels, since cos C is a ratio, which is exactly what the absence of numbers predicted. The second idea is to stop looking at △ ABC and look instead at the small triangles that share the vertex G. The equilateral condition puts a known 60° at G, and because both medians pass straight through G, the other angles there are forced to 120° and 60° by supplementary and vertical pairs. Each small triangle then has two known sides and the angle between them, which is precisely the Law of Cosines setup. Two applications produce AB and BC; a third application, this time in the big triangle at vertex C, produces cos C.
Name the equilateral side
Give the side a name.
When a problem gives shape but no size, you are allowed to choose the size, and the answer must not care which choice you made.
10.G-CO.C.10Introduce A VariableUnfold with the two-to-one rule
Each median splits two to one.
The centroid always splits a median into a long piece and a short piece in a 2:1 ratio, so knowing any one piece hands you the whole median.
The centroid splits a median into a long piece and a short piece in a fixed two-to-one ratio.
▸ Why?
The medians cut out triangles with the same angles, so their matching sides sit in one fixed ratio.
▸ Why?
That ratio cuts the median into three equal shares, so knowing one piece hands over the whole.
Read the angles at the crossing
Vertical angles fix the angles.
Two straight lines crossing at a point make only two distinct angles, so one known angle at G tells you all four.
10.G-CO.C.9Draw A DiagramFind the first side
The law of cosines gives one side.
Two sides with a known angle wedged between them already determine the third side, and the Law of Cosines is the machine that reads it off.
11.G-SRT.D.11Convert To AlgebraFind the second side
It gives another side too.
The midpoint D is the bridge: solve the small triangle that touches it, then double to cross back to the full side.
11.G-SRT.D.11Convert To AlgebraGet the cosine from three sides
Collect all three and take the cosine.
Once all three sides are known the triangle is rigid, so every angle in it is already decided and the Law of Cosines just reports the value.
11.G-SRT.D.11Organize Information In More WaysRationalize and add
Rationalizing and adding gives 44.
A required form is part of the question, so the last job is bookkeeping: push the radical up top, reduce, and only then read the integers.
11.N-RN.A.2Introduce A VariableWhen a shape problem gives no numbers, name one length yourself and let the fixed rules — a midpoint doubles, a centroid splits 2:1, a straight line makes 180° — carry that name to every other length.
- Name the equilateral side
- Unfold both medians with the 2 to 1 rule
- Read the angles around G
- Law of Cosines in triangle ABG gives AB
- Law of Cosines in triangle BGD gives BC
- All three sides, then Law of Cosines at C
- Rationalize, then read off m, n, p
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