Problem 26 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
area-decompositionfolding
Vilma took a sheet of paper measuring 10 cm × 20 cm and made two folds, bringing the two shorter sides onto a diagonal of the sheet. She obtains a parallelogram, as shown. What is the area of this quadrilateral, in cm²?
Show answer
Answer: D — \(50(5 - \sqrt{5})\)
Show hints
Hint 1 of 2
Each fold turns a short side onto the diagonal along an angle-bisector crease.
Still stuck? Show hint 2 →
Hint 2 of 2
Find where each crease meets a long edge, then subtract the two folded triangles.
Show solution
Approach: locate the bisector creases and subtract the folded triangles
- Folding a short side onto the diagonal creases along the bisector of the corner angle.
- That crease meets a long edge a distance \(5(\sqrt{5}-1)\) from the corner, so each folded triangle has area \(\tfrac12 \cdot 10 \cdot 5(\sqrt{5}-1) = 25(\sqrt{5}-1)\).
- Parallelogram area \(= 200 - 2 \cdot 25(\sqrt{5}-1) = 250 - 50\sqrt{5} = 50(5-\sqrt{5})\) cm², option D.
Mark:
· log in to save