Let AB be a diameter of a circle, and let C be a point on the circle such that AC = 8 and BC = 8. The angle bisector of angle ACB intersects the circle at point M. Find CM.
Use the angle bisector theorem. The angle bisector theorem states that the ratio of the lengths of two segments that are bisected by an angle bisector is equal to the ratio of the lengths of the other two segments that are bisected by the angle bisector. In this case, the two segments that are bisected by the angle bisector are AC and BC. The other two segments are AM and CM. So, the ratio of AM to CM is equal to the ratio of AC to BC.
Find the length of AM. The length of AM can be found using the Pythagorean theorem. AB is a diameter of the circle, so AB = 10. The length of AC is 8, so the length of AM is equal to 6. The ratio of AM to CM is equal to the ratio of AC to BC, which is 1:1. So, the length of CM is equal to 6.
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