Problem 23 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
A square with area 30 is split into two by its diagonal and then split into triangles as shown in the diagram. Some of the areas of the triangles are given in the diagram. Which of the line segments a, b, c, d, e of the diagonal is the longest?
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Answer: D — d
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Hint 1 of 2
Triangles on the diagonal share the same height, so a segment's length is proportional to its triangle's area.
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Hint 2 of 2
Read each segment a, b, c, d, e from the area sitting on it and pick the biggest.
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Approach: segment length is proportional to the triangle's area
- The triangles standing on the diagonal share one height, so each base segment is proportional to that triangle's area.
- The square area 30 is split into the marked pieces (5, 9, 4, 2 and the rest), letting each segment be read off its area.
- Comparing all five, segment d carries the largest area and so is the longest.
- So the longest segment is d (D).
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