Multiple choice

State the following statement is true or false The measure of the line segment joining the centre of a circle to the mid-point of a chord is $\dfrac{\sqrt{4r^{2}-L^{2}}}{2}$

  1. True

  2. False
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A Correct answer
Explanation

The line segment from the center of a circle to the midpoint of a chord is perpendicular to the chord, forming a right triangle. The hypotenuse of this triangle is the radius r, and one leg is half the length of the chord, L/2. Applying the Pythagorean theorem, the distance d satisfies d^2 + (L/2)^2 = r^2, which simplifies to d = sqrt(4r^2 - L^2) / 2.

AI explanation

According to the perpendicular bisector property of a circle, the line drawn from the center of a circle perpendicular to a chord bisects the chord. This forms a right triangle where the radius r is the hypotenuse, half the chord length L/2 is one leg, and the distance from the center to the midpoint is the other leg. By the Pythagorean theorem, the distance equals the square root of r^2 - (L/2)^2, which simplifies to the square root of (4r^2 - L^2) / 2. The statement is true.

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