# A triangle ABC angle C = 90 degrees Hypotenuse AB is divided by the height СH. Drawn from the vertex

A triangle ABC angle C = 90 degrees Hypotenuse AB is divided by the height СH. Drawn from the vertex of the right angle into segments AH-18cm and BH-8cm. Find the height CH.

The tangent of an angle in a right-angled triangle is equal to the ratio of the lengths of the opposite to the adjacent leg.

Consider triangle AHC. In it we find the tangent of angle A:

tg (A) = CH / AH.

Let’s find what the angle BCH is equal to:

∠BCH = 90 ° – ∠ABC = 90 ° – (90 ° – ∠A) = ∠A.

Consider a triangle BCH. Find the tangent of the angle BCH:

tg (∠BCH) = BH / CH.

If the angles are equal, then the tangents are also equal. Equating the tangents, we get:

CH / AH = BH / CH;

CH2 = AH * BH;

CH = √ (AH * BH);

CH = √ (18 * 8);

CH = 12.

Answer: the height is 12 cm.

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