Three pairwise intersecting chords of equal length are drawn in the circle. Each chord is divided by intersection

Three pairwise intersecting chords of equal length are drawn in the circle. Each chord is divided by intersection points into three parts of equal length. Find the radius of the circle if the length of each of the chords is a.

Since all chords are equal and are divided at the points of intersection into equal segments, the triangle formed by the points of their intersection is equilateral, with side (a / 3) cm. Point O is the center of the circle, the center of the triangle and the center of the circle, which can be inscribed into it …

OH = (a / 3) / 2 * √3 = a / 6 * √3 = a * √3 / 18 cm.

Triangle AOН is rectangular, in which the leg AH = (a / 3) + (a / 6) = a / 2 cm.

Then, by the Pythagorean theorem: OA ^ 2 = OH ^ 2 + AH ^ 2 = (a ^ 2/4) + (3 * a ^ 2/324) = 84 * a ^ 2/324.

ОА = R = √ (21 * a ^ 2/81) = a * √21 / 9 cm.

Answer: The radius of the circle is a * √21 / 9 cm.