Problem 11 · 2013 Math Kangaroo
Hard
Geometry & Measurement
transformationssymmetry
Triangle RZT is generated by rotating the equilateral triangle AZC about point Z. Angle \(\beta = \angle CZR = 70^\circ\). Determine angle \(\alpha = \angle CAR\).
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Answer: D — \(35^\circ\)
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Hint 1 of 2
A rotation about Z keeps lengths, so ZC = ZR and triangle CZR is isosceles.
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Hint 2 of 2
Find the base angle of that isosceles triangle, then relate it to angle CAR.
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Approach: use the rotation to build an isosceles triangle
- Rotating equilateral triangle AZC about Z sends C to R, so ZC = ZR.
- Triangle CZR is isosceles with apex angle β = 70°, so its base angles are (180° − 70°) / 2 = 55°.
- Combining the 60° angle of the equilateral triangle at A with this geometry gives α = 35°.
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