Problem 22 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
reflectionpythagorean-triple
A quadrilateral ABCD has two right angles, at the vertices B and C. It is known that AB = 4, BC = 8 and CD = 2. What is the smallest possible value of AX + DX, if X is a point on the segment BC?
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Answer: D — 10
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Hint 1 of 2
Reflect one of the right-angle vertices across line BC so the path AX + DX becomes straight.
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Hint 2 of 2
After reflecting A across BC, the shortest AX + DX is the straight distance to D — a right-triangle hypotenuse.
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Approach: reflect to straighten a shortest path
- With right angles at B and C, AB = 4 and CD = 2 stand perpendicular to BC = 8.
- Reflect A across line BC to get A'; then AX + DX = A'X + DX, smallest when A', X, D are collinear.
- That distance is √(8² + (4+2)²) = √100 = 10.
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