Problem 9 · 2024 Math Kangaroo
Medium
Geometry & Measurement
symmetry
A square has vertices A, B, C, D as shown, and a regular hexagon is drawn on the side OC, where O is the centre of the square. How big is the angle α?
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Answer: A — 105°
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Hint 1 of 2
O is the centre, so OC is half a diagonal of the square; that makes triangle OCD a nice special triangle.
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Hint 2 of 2
A regular hexagon has interior angles of 120 degrees, so the hexagon edge leaving O is tilted a fixed amount from OC.
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Approach: find the tilt of the hexagon edge, then subtract from the straight side
- Since O is the square's centre, OC and OD are both half-diagonals and CD is a side, so triangle OCD is right-angled and isosceles with the angle at C equal to 45 degrees and OC making 45 degrees with the horizontal top side AD.
- At O the hexagon's interior angle is 120 degrees, so the hexagon edge leaving O is turned 120 degrees from OC, which lands it pointing 75 degrees above the horizontal.
- The angle \(\alpha\) between that rising hexagon edge and the horizontal side AD is \(180^\circ-75^\circ=105^\circ\), answer A.
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