Problem 12 · 2025 Math Kangaroo
Medium
Geometry & Measurement
foldingreflection
Alexander folds a square sheet of paper along its diagonal to form a triangle. He then folds the paper again so that one of the two shorter sides of the triangle lies on the longer side of the triangle to form the smaller triangle AXC (see diagrams). What is the size of the angle ∠CXA?
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Answer: B — 112.5°
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Hint 1 of 2
After the first fold you have a right isosceles triangle, with angles 90°, 45°, 45°.
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Hint 2 of 2
The second fold lays a 45° leg onto the hypotenuse, so the crease bisects the angle it folds — work out the resulting angle at X.
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Approach: track angles through the two folds
- Folding the square along its diagonal gives a right isosceles triangle: a 90° corner and two 45° corners.
- The second fold lays a short side (leg) onto the long side (hypotenuse), so the crease through X bisects the 45° corner at A into two 22.5° pieces.
- At X the crease meets the right angle, giving \(\angle CXA = 90^\circ + 22.5^\circ = 112.5^\circ\), answer B.
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