Problem 21 · 2012 Math Kangaroo
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Geometry & Measurement
casework
Two sides of a quadrilateral have lengths 1 and 4. One of the diagonals has length 2 and splits the quadrilateral into two isosceles triangles. What is the perimeter of the quadrilateral?
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Answer: D — 11
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Hint 1 of 2
The diagonal of length 2 makes each of the two triangles isosceles.
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Hint 2 of 2
Pair the given sides 1 and 4 with the diagonal so each triangle's two equal sides work out.
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Approach: force each triangle to be isosceles using the diagonal
- The diagonal of length 2 splits the quadrilateral into two isosceles triangles.
- On the triangle holding the side 1, the only valid isosceles choice is sides 1, 2, 2; on the triangle holding the side 4 it is 4, 4, 2 (sides 2, 2, 4 would be degenerate).
- So the four sides are 1, 2, 4, 4, giving perimeter 1 + 2 + 4 + 4 = 11 (D).
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