Problem 9 · 2005 AMC 8
Medium
Geometry & Measurement
isosceles-then-equilateral
In quadrilateral ABCD, sides AB and BC both have length 10, sides CD and DA both have length 17, and the measure of angle ADC is 60°. What is the length of diagonal AC?
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Answer: D — 17.
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Hint 1 of 2
The 10-10 sides are a distraction. The diagonal AC lives inside triangle ADC, which has two equal sides (17, 17) and the 60° angle between them. That's all you need.
Still stuck? Show hint 2 →
Hint 2 of 2
Watch what happens to an isosceles triangle when the apex angle is exactly 60° — check what the other two angles must be.
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Approach: isosceles + a 60° angle forces equilateral
- Look only at triangle ADC: DA = DC = 17, so its base angles are equal. They share the leftover 180 − 60 = 120°, giving 60° each.
- All three angles are 60° — the triangle is equilateral, so every side equals 17. Thus AC = 17.
- Worth keeping: isosceles + one 60° angle = equilateral, every time. Spotting it beats hauling out the Law of Cosines. (Choice D, 17, is no coincidence — AC matches the equal sides.)
Another way — Law of Cosines (the unenlightened route):
- AC² = 17² + 17² − 2·17·17·cos 60°. Since cos 60° = ½, the last term is 17², leaving AC² = 17².
- So AC = 17 — the same result, the long way. The cos 60° = ½ collapse is exactly why the triangle came out equilateral.
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