In triangle ABC, angle C is 90 degrees, BC = √135. The radius of the circle around this triangle is 8. Find AC.

Since the circle is circumscribed about a right-angled triangle, the hypotenuse AB of the right-angled triangle ABC coincides with the diameter of the circle.

Then the length of the hypotenuse AB is equal to two lengths of the radius of the circle. AB = 2 * R = 2 * OA = 2 * 8 = 16 cm.

In a right-angled triangle ABC, according to the Pythagorean theorem, we determine the length of the leg AC.

AC ^ 2 = AB ^ 2 – BC ^ 2 = 16 ^ 2 – (√135) ^ 2 = 256 – 135 = 121.

AC = √121 = 11 cm.

Answer: The length of the speaker side is 11 cm.