Numerical answer
In triangle ABC, AB = AC = 10 cm and BC = 12 cm. Find the radius (in cm) of the circle passing through A, B and C.
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Answer: 6.25
The altitude from A to BC bisects BC, so height = √(10² − 6²) = 8 cm and area = ½ × 12 × 8 = 48 cm².
Circumradius R = (abc)/(4 × Area) = (10 × 10 × 12)/(4 × 48) = 1200/192 = 6.25 cm.
Alternative: The centre lies on the altitude at distance R from A: (8 − R)² + 6² = R² → 64 − 16R + 36 = 0 → R = 100/16 = 6.25.
Circumradius R = (abc)/(4 × Area) = (10 × 10 × 12)/(4 × 48) = 1200/192 = 6.25 cm.
Alternative: The centre lies on the altitude at distance R from A: (8 − R)² + 6² = R² → 64 − 16R + 36 = 0 → R = 100/16 = 6.25.
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