Diameter AB and chord AC are drawn through point A lying on the circle, with AC = 10

Diameter AB and chord AC are drawn through point A lying on the circle, with AC = 10, angle BAC = 60 (degrees). Find the length of the chord CM perpendicular to AB.

Let’s draw from point C a chord CM, perpendicular to the diameter AB of the circle.

Since the diameter of the circle perpendicular to the chord divides the chord into equal parts, then СD = MD.

Consider a right-angled triangle AСD, in which angle D is straight, angle A = 30 by condition, hypotenuse AC = 8 cm.Since the leg of the СD lies opposite angle 30, its length is equal to half the length of the hypotenuse, therefore, СD = AC / 2 = 8 / 2 = 4 cm.

Then MD = СD = 4 cm, and СM = MD + СD = 4 + 4 = 8 cm.

Answer: The length of the CM chord is 8 cm.