Problem 15 · 2021 Math Kangaroo
Medium
Geometry & Measurement
spatial-reasoning
Five identical right-angled triangles can be arranged so that their larger acute angles touch to form the star shown in the diagram. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch. How many triangles are needed to form the second star?
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Answer: D — 20
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Hint 1 of 2
Five large acute angles meeting at a point fill 360°, so each large acute angle is 72°.
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Hint 2 of 2
The small acute angle is 90° minus the large one; see how many fill a full turn.
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Approach: use the angles meeting at the star's centre
- For the first star, 5 equal larger acute angles surround the centre: 360° ÷ 5 = 72°.
- The triangle is right-angled, so the smaller acute angle is 90° − 72° = 18°.
- For the second star, smaller angles meet at the centre: 360° ÷ 18° = 20 triangles.
- So the answer is D.
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