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2025 Math Kangaroo

Problem 10

Problem 10 · 2025 Math Kangaroo Medium
Geometry & Measurement square-areaarea-fraction

Consider a circle with centre O and radius 10 cm. A square OPQR is drawn inside the circle so that Q lies on the circle (see diagram). What is the area of the grey triangle PQR?

Figure for Math Kangaroo 2025 Problem 10
Show answer
Answer: B — 25 cm²
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Hint 1 of 2
The corner of the square at the centre and the opposite corner on the circle are the ends of a diagonal.
Still stuck? Show hint 2 →
Hint 2 of 2
The diagonal equals the radius; the triangle is half the square.
Show solution
Approach: diagonal = radius, triangle is half the square
  1. O is a corner of square OPQR and Q is the opposite corner on the circle, so the diagonal OQ = radius = 10.
  2. A square with diagonal d has area d²/2 = 100/2 = 50.
  3. Triangle PQR is half the square = 25 cm².
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