Multiple choice

The radius of a circle is $16cm$. The mid point of a chord of the circle lies on the diameter perpendicular to the chord and its distance from the near end of the diameter is $3cm$; If the length of that chord is $m \sqrt{87}$ cm, then the value of $m$ is

  1. $4$
  2. $8$
  3. $2$
  4. $1$
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C Correct answer
Explanation

Let the diameter be AB. The chord is perpendicular to the diameter at a point P. Distance from near end is 3, so distance from center is 16-3 = 13. In the right triangle formed by radius (16), distance to center (13), and half-chord (h), h^2 = 16^2 - 13^2 = 256 - 169 = 87. Half-chord = sqrt(87). Full chord = 2*sqrt(87). So m=2.

AI explanation

The diameter perpendicular to the chord bisects the chord, so we use the Pythagorean theorem on the right triangle formed by the radius, the distance to the chord, and half the chord. The distance from the center to the chord is 16 minus 3, which equals 13. Squaring the radius 16 gives 256, and squaring the distance 13 gives 169; their difference is 87. The square root of this difference is the square root of 87, which equals half the chord length. Since the full chord is m times the square root of 87, m equals 2.

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