Problem 20 · 2014 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triplesymmetry
PQRS is a rectangle. T is the midpoint of RS. QT is perpendicular to the diagonal PR. What is the ratio of the lengths PQ : QR?
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Answer: D — \(\sqrt{2}:1\)
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Hint 1 of 2
Put the rectangle on coordinates and write QT and the diagonal PR as vectors.
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Hint 2 of 2
Perpendicular means their dot product is zero.
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Approach: coordinates and a perpendicularity (dot-product) condition
- Let P=(0,0), Q=(a,0), R=(a,b), S=(0,b); then T, the midpoint of RS, is (a/2, b).
- PR has direction (a,b) and QT has direction (−a/2, b); perpendicular means −a²/2 + b² = 0.
- So a² = 2b², giving a/b = √2.
- Hence PQ : QR = √2 : 1.
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