Multiple choice

A chord of length $L$ cm is drawn in a circle of radius $R$ cm. The distance of the chord from the center of the circle is

  1. $\displaystyle \sqrt{L^{2}}-\frac{R^{2}}{4}$
  2. $\displaystyle \sqrt{\frac{4R^{2}-L^{2}}{4}}$
  3. $\displaystyle \beta $
  4. $\displaystyle \sqrt{R^{2}}+L^{2}-RL$
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B Correct answer
Explanation

In a circle, the radius, half-chord, and distance from center form a right triangle. R^2 = d^2 + (L/2)^2. Thus, d^2 = R^2 - L^2/4 = (4R^2 - L^2)/4. d = sqrt((4R^2 - L^2)/4).

AI explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord, creating a right triangle with the radius as the hypotenuse, half the chord length as one leg, and the distance from the center as the other leg. Using the Pythagorean theorem, the distance d squared equals R squared minus (L/2) squared. Simplifying this expression by finding a common denominator yields d squared equals (4R squared minus L squared) divided by 4. Taking the square root gives the distance as the square root of ((4R squared minus L squared) divided by 4).

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