# The height BM of triangle ABC divides its side AC into segments AM and CM. Find the length

The height BM of triangle ABC divides its side AC into segments AM and CM. Find the length of the segment CM, if AB = 12√3, BC = 20 cm, angle A = 45 degrees.

1. We calculate the length of the BM height through the sine ∠A = 90 °. Sine ∠A is the quotient of dividing the height of the BM, which in a right-angled triangle ABM is a leg by the length of the hypotenuse AB.

BM: AB = sine ∠A = sine 45 ° = √2 / 2.

BM = AB x √2 / 2 = 12√2 x √2 / 2 = 12 centimeters.

2. We calculate the length of the required segment SM (by the Pythagorean theorem):

CM = √BC²- BM² = √20²- 12² = √400 – 144 = √256 = 16 centimeters.

Answer: The length of the CM segment is 16 centimeters.

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