Problem 23 · 2014 Math Kangaroo
Stretch
Geometry & Measurement
areaproportion
The quadrilateral ABCD has right angles only at corners A and D. The numbers in the diagram give the areas of the triangles in which they are located. What is the area of ABCD?
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Answer: B — 45
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Hint 1 of 2
Right angles at A and D make AB and DC both perpendicular to AD, so AB is parallel to DC — ABCD is a trapezoid.
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Hint 2 of 2
In a trapezoid the two diagonals cut it into four triangles; the two 'side' triangles (on the legs) always have equal area, and the top and bottom triangles are similar.
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Approach: use the trapezoid-diagonal area relations
- Since the angles at A and D are right angles, AB and DC are both perpendicular to AD, so AB is parallel to DC and ABCD is a trapezoid with diagonals AC and DB.
- The two triangles on the legs are equal in area, so the triangle on the right (T) equals the given left triangle: T = 10.
- The top triangle (5) and bottom triangle (S) are similar, and a diagonal trapezoid gives \(10^2 = 5 \times S\), so S = 20.
- Adding all four triangles: 5 + 10 + 10 + 20 = 45.
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