Find the length of leg AB in right-angled triangle ABC if it is known that the ratio: AC: AB = 8: 6, and the hypotenuse BC = 20.

univerkov | February 20, 2021 | education

First way.

Let the length of the segment AB = 6 * X cm, then, by condition, the length of the segment AC = 8 * X cm.

By the Pythagorean theorem, BC ^ 2 = AC ^ 2 + BC ^ 2.

400 = 64 * X ^ 2 + 36 * X ^ 2 = 100 * X ^ 2.

X ^ 2 = 400/100 = 4.

X = 2.

Then AB = 2 * 6 = 12 cm.

Second way.

Since AC / AB = 8/6, then tgABC = 8/6 = 4/3.

Then Cos2ABC = 1 / (1 + tg2ABC) = 1 / (1 + 16/9) = 9/25.

CosABC = 3/5.

Then AB = BC * СosABC = 20 * 3/5 = 12 cm.

Answer: The length of the AB side is 12 cm.

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