Multiple choice

Directions: There is a circular field with radius 6.9 cm and length of its chord is 9.75 cm. Find the appoximate angle that the chord subtends at centre of the circle.

  1. 60°

  2. 90°

  3. 30°

  4. 120°

  5. 45°
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B Correct answer
Explanation

Using the law of cosines: c^2 = a^2 + b^2 - 2ab*cos(theta). 9.75^2 = 6.9^2 + 6.9^2 - 2 * 6.9 * 6.9 * cos(theta). 95.0625 = 47.61 + 47.61 - 95.22 * cos(theta). 95.0625 = 95.22 - 95.22 * cos(theta). 95.22 * cos(theta) = 0.1575. cos(theta) is near 0, so theta is approximately 90 degrees.

AI explanation

The perpendicular distance from the center to the chord is calculated using the Pythagorean theorem: d equals the square root of (r squared minus half the chord length squared). This gives the square root of (6.9 squared minus 4.875 squared), which is the square root of (47.61 minus 23.76), equaling approximately 4.88 cm. This perpendicular distance is very close to the radius of 6.9 cm, making the right triangle inside nearly isosceles, so the half angle is roughly 45 degrees and the total angle subtended at the center is exactly 90 degrees.

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