Problem 19 · 2017 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triple
In a convex quadrilateral ABCD the diagonals are perpendicular to each other. The lengths of the edges are AB = 2017, BC = 2018 and CD = 2019 (diagram not to scale). How long is side AD?
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Answer: D — \(\sqrt{2018^2 + 2}\)
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Hint 1 of 2
With perpendicular diagonals, the four sides satisfy a neat relation between opposite pairs.
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Hint 2 of 2
Use AB^2 + CD^2 = BC^2 + AD^2 to solve for AD.
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Approach: apply the perpendicular-diagonal identity for the sides
- For perpendicular diagonals, AB^2 + CD^2 = BC^2 + AD^2.
- So AD^2 = 2017^2 + 2019^2 - 2018^2.
- Since 2017^2 + 2019^2 = 2*2018^2 + 2, this gives AD^2 = 2018^2 + 2, so AD = sqrt(2018^2 + 2).
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