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2022 Math Kangaroo

Problem 26

Problem 26 · 2022 Math Kangaroo Stretch
Geometry & Measurement areaarea-decomposition

The diagonals of the squares ABCD and EFGB are 7 cm and 10 cm long respectively (see diagram). The point P is the point of intersection of the two diagonals of the square ABCD. How big is the area of the triangle FPD (in cm²)?

Figure for Math Kangaroo 2022 Problem 26
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Answer: E — 17.5
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Hint 1 of 2
Place the squares on coordinates using their diagonals 7 and 10 sharing vertex B.
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Hint 2 of 2
P is the centre of square ABCD; find F, P, D coordinates and take the triangle area.
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Approach: coordinate geometry from the shared vertex and diagonals
  1. Set coordinates so the squares ABCD (diagonal 7) and EFGB (diagonal 10) share the vertex B.
  2. Locate F, P (centre of ABCD) and D from the side lengths derived from the diagonals.
  3. The area of triangle FPD computes to 17.5.
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