Multiple choice

In a triangle ABC, AB = 12cm and BC = 16cm. If circumcentre O of the triangle ABC lies on the line segment AC then what is the area of triangle AOB?

  1. 96

  2. 72

  3. 48

  4. cannot be determined
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If the circumcentre lies on AC, then AC is the diameter of the circle. Triangle ABC is a right-angled triangle at B. Area = 0.5 * AB * BC = 0.5 * 12 * 16 = 96. Since O is the midpoint of AC, triangle AOB has half the area of triangle ABC? No, O is the midpoint of the hypotenuse, so AO = OC = OB (radius). Triangle AOB is isosceles with AO=OB. The altitude from O to AB is half of BC = 8. Area = 0.5 * 12 * 8 = 48.

AI explanation

Because the circumcentre lies on the side AC, triangle ABC is a right-angled triangle with the right angle at B and hypotenuse AC. The circumradius is half the hypotenuse, so AC equals 20 cm using the Pythagorean theorem with sides AB as 12 cm and BC as 16 cm. The circumradius OA is 10 cm, and drawing the altitude from O to AB in triangle AOB gives an area of 48 square cm.

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