Problem 7 · 2010 Math Kangaroo
Medium
Geometry & Measurement
symmetry
The triangle pictured is right-angled. M is the midpoint of the hypotenuse AB and ∠BCA = 90°. How big is ∠BMC?
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Answer: D — 120°
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Hint 1 of 2
The midpoint of a right triangle's hypotenuse is the same distance from all three vertices.
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Hint 2 of 2
That makes triangle BMC isosceles; use the marked angle.
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Approach: midpoint of hypotenuse = circumcentre
- Since M is the midpoint of the hypotenuse AB and the angle at C is 90°, MB = MC = MA.
- So triangle BMC is isosceles with MB = MC, giving equal base angles.
- With the angle at A = 60°, the angle at B = 30°, so ∠MBC = 30°.
- Then ∠BMC = 180° − 30° − 30° = 120°.
Mark:
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