Question : A circle with centre O has a chord AB that is 20 cm in length. If the radius of the circle is 12 cm, then the area of triangle AOB is:
Option 1: $20 \sqrt{15}$ cm2
Option 2: $22 \sqrt{11}$ cm2
Option 3: $20 \sqrt{11}$ cm2
Option 4: $22 \sqrt{15}$ cm2
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Correct Answer: $20 \sqrt{11}$ cm 2
Solution :
The area of triangle AOB,
$ = \frac{1}{2} \times \text{base} \times \text{height}$
In triangle AOB, AB is the base and the height is the perpendicular drawn from the centre O to the chord AB.
Use the Pythagorean theorem,
$OD = \sqrt{12^2 - 10^2} = \sqrt{44} = 2\sqrt{11}$ cm
The area of triangle AOB,
$\text{Area} = \frac{1}{2} \times AB \times OD = \frac{1}{2} \times 20 \times 2\sqrt{11} = 20\sqrt{11}$ cm
2
Hence, the correct answer is $20\sqrt{11}$ cm
2
.
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