Problem 30 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
Consider the pentagon \(ABCDE\) with \(\angle BAE = \angle CBA = 90^\circ\), \(\overline{AE} = \overline{BC}\) and \(\overline{ED} = \overline{DC}\). Four points are marked along \(AB\), dividing it into five pieces of equal length, and vertical lines are drawn through these points as shown in the diagram. The dark part in the middle has an area of 13 cm² and the lightly shaded part to its left has an area of 10 cm². What is the area of the entire pentagon, in cm²?
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Answer: A — 45
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Hint 1 of 3
The pentagon is a vertical-sided base with a triangular roof peaking at D, and it is symmetric about the line through D, so the five strips pair up: 1st = 5th and 2nd = 4th.
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Hint 2 of 3
Each strip's area equals its (equal) width times its average height, and along a straight roof edge the average height climbs by the same fixed amount from one strip to the next.
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Hint 3 of 3
Use the two given strips to pin down that climb and the lowest strip, then add all five.
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Approach: use the straight roof edges to find the side strip, then sum the five symmetric strips
- Along the left roof edge the strips' average heights rise by a constant step, so the 1st and 2nd strips differ by the same amount the slope adds per strip; the middle (3rd) strip straddles the peak.
- Matching the given 2nd strip = 10 and middle strip = 13 fixes the geometry, which makes the 1st (and by symmetry 5th) strip equal to 6.
- The five strips are therefore 6, 10, 13, 10, 6.
- Their total is 6 + 10 + 13 + 10 + 6 = 45 cm².
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