Problem 20 · 2021 Math Kangaroo
Hard
Geometry & Measurement
areasquare-area
The diagram shows three squares, PQRS, TUVR and UWXY. They are placed together, edge to edge. Points P, T and X lie on the same straight line. The area of PQRS is 36 and the area of TUVR is 16. What is the area of triangle PXV?
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Answer: C — \(16\)
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Hint 1 of 2
Set coordinates with the shared baseline on the x-axis and read off the corner points.
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Hint 2 of 2
Use the collinearity of P, T, X to find the small square's size, then apply the shoelace area.
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Approach: coordinatise and use the shoelace formula
- Place S=(0,0): then P=(0,6), R=(6,0), V=(10,0), T=(6,4), U=(10,4) from the side-6 and side-4 squares.
- P, T, X collinear (line y = 6 − x/3) forces the third square's side 2, giving X=(12,2).
- Shoelace on P(0,6), X(12,2), V(10,0): area = ½|0−72+40| = 16.
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