Problem 12 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
Two equilateral triangles of different sizes are placed on top of each other so that a hexagon is formed on the inside whose opposite sides are parallel. Four of the side lengths of the hexagon are stated in the diagram. How big is the perimeter of the hexagon?
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Answer: D — 70
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Hint 1 of 2
Two equilateral triangles overlapping make a hexagon whose every interior angle is 120°.
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Hint 2 of 2
In an equiangular hexagon the differences of opposite sides are all equal.
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Approach: use the equiangular-hexagon side relation
- Because the hexagon is formed by two equilateral triangles, all six angles are 120°.
- For an equiangular hexagon with sides in order, opposite-side differences are all equal; the known sides 6, 15, 11, 12 then give the missing sides 9 and 17.
- Perimeter = 6+15+11+12+9+17 = 70.
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