Multiple choice

Two congruent circles of radius r intersect such that each passes through the centre of the other, then the length of the common chord is given by ?

  1. $r$
  2. $2r$
  3. $\sqrt 2r$
  4. $\sqrt 3r$
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D Correct answer
Explanation

Two circles of radius r intersect such that each passes through the center of the other. The distance between centers is r. The common chord forms two equilateral triangles with the centers, each with side r. The height of one triangle is (sqrt(3)/2) * r. The total length of the chord is 2 * (sqrt(3)/2) * r = sqrt(3) * r.

AI explanation

The common chord and the two radii form two equilateral triangles of side r, meaning the line connecting the centers is the perpendicular bisector of the chord. Using the Pythagorean theorem in one of the right triangles formed, half the chord length is the square root of r squared minus r over 2 squared, which equals r root 3 over 2. The full length of the common chord is double this value, resulting in root 3 r.

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