Multiple choice

A chord of length 12 cm is drawn in a circle of radius 10 cm. A triangle is formed by joining the ends of the chord to the centre. What is the area of the triangle?

  1. 24 sq. cm

  2. 36 sq. cm

  3. 48 sq. cm

  4. 60 sq. cm

  5. 120 sq. cm
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C Correct answer
Explanation

The triangle formed by the chord and the center is an isosceles triangle with sides 10, 10, and 12. By drawing an altitude to the chord, we split it into two right triangles with hypotenuse 10 and base 6, making the height 8 via the Pythagorean theorem. The area is 1/2 * base * height = 1/2 * 12 * 8 = 48 sq. cm.

AI explanation

The perpendicular drawn from the center of the circle to the chord bisects the chord, creating a right triangle with the radius of 10 cm as the hypotenuse and half the chord length of 6 cm as one leg. Using the Pythagorean theorem, the distance from the center to the chord is calculated as sqrt(10^2 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 cm, which serves as the height of the formed isosceles triangle. The area of the triangle is then 0.5 * base * height = 0.5 * 12 * 8 = 48 sq. cm. The area is 48 sq. cm.

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