Problem 24 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
area-fractionratio
Lines parallel to the base AC of triangle ABC are drawn through X and Y. In each case, the areas of the grey parts are equal in size. The ratio BX : XA = 4 : 1 is known. What is the ratio BY : YA?
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Answer: D — 3 : 2
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Hint 1 of 2
A line parallel to the base cuts off a small similar triangle whose area scales as the square of the height ratio.
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Hint 2 of 2
Set the two grey areas equal: (BY/BA)² matches 1 − (BX/BA)², with BX/BA = 4/5.
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Approach: use area scales as the square of the side ratio
- BX : XA = 4 : 1 means BX/BA = 4/5, so the trapezoid below X is 1 − (4/5)² = 9/25 of the triangle.
- On the right, the grey triangle above Y is (BY/BA)² of the triangle.
- Setting (BY/BA)² = 9/25 gives BY/BA = 3/5, so BY : YA = 3 : 2.
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