Multiple choice

O is the circumcentre of a triangle ABC with circumradius 13 cm. If BC = 24 cm and OD is perpendicular to BC, then the length of OD is

  1. 3 cm

  2. 4 cm

  3. 5 cm

  4. 7 cm
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C Correct answer
Explanation

Since O is the circumcentre of triangle ABC, the distance from O to any vertex is equal to the circumradius, so OB = 13 cm. The perpendicular OD from the center O to the chord BC bisects the chord, meaning BD = 12 cm. Applying the Pythagorean theorem in the right-angled triangle ODB, we get OD^2 + 12^2 = 13^2, which simplifies to OD = 5 cm.

AI explanation

A line drawn from the circumcentre perpendicular to a chord bisects the chord, meaning BD is half of BC, giving 12 cm. This creates a right-angled triangle OBD where the hypotenuse is the circumradius OB at 13 cm and one leg is BD at 12 cm. Applying the Pythagorean theorem, OD squared equals OB squared minus BD squared, so OD squared equals 169 minus 144, resulting in OD equal to 5 cm.

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