Problem 26 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
proportionarea
In a rectangle JKLM the angle bisector at J intersects the diagonal KM at N. The distance of N to LM is 1 and the distance of N to KL is 8. How long is LM?
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Answer: A — \(8+2\sqrt{2}\)
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Hint 1 of 2
The bisector from J makes equal angles, so drop the two given distances as the legs of similar right triangles.
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Hint 2 of 2
Combine the distance-to-LM = 1 and distance-to-KL = 8 conditions with where N sits on diagonal KM.
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Approach: place coordinates and use the bisector’s 45° line
- Put M at the origin; the bisector from J runs at 45°, so N’s drop to the base and its horizontal match along that line.
- The conditions (height of N = 1, distance to side KL = 8) give (h − 1)² = 8 for the rectangle’s height.
- Solving for the base length LM yields 8 + 2√2.
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