Problem 18 · 2011 Math Kangaroo
Hard
Geometry & Measurement
areacasework
Louise draws a line DE of length 2 cm. How many ways are there for her to add a point F so that a right-angled triangle DEF with area 1 cm² can be formed?
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Answer: C — 6
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Hint 1 of 2
Area 1 with base 2 forces the distance from F to line DE to be 1.
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Hint 2 of 2
Consider separately where the right angle is: at D, at E, or at F.
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Approach: place the right angle at D, at E, or at F
- With base DE = 2 and area 1, the height must be 1.
- Right angle at D: F is 1 cm from D perpendicular to DE — 2 positions (each side); same at E gives 2 more.
- Right angle at F: F lies on the circle with diameter DE, and height 1 is reached at exactly 2 points.
- Total 2+2+2 = 6 positions.
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