# Right-angled triangle KMN. Angle M = 90 degrees. On the side КN from the point М the height МН is lowered.

Right-angled triangle KMN. Angle M = 90 degrees. On the side КN from the point М the height МН is lowered. KH = 25cm, NH = 144cm. Find: NM, KM and NM.

To solve the problem, we will apply the property of the height of a right-angled triangle.

Let’s remember it: the height drawn from the vertex of the right angle is the geometric mean for the segments of the hypotenuse into which it is divided by this height.

Find the length MH:

МН = √ (KH * HN) = √ (25 * 144) = √25 * √144 = 12 * 5 = 60 cm.

We use the property of the leg of a right-angled triangle.

Let’s remember it: the leg of a right-angled triangle is the geometric mean for the hypotenuse and the segment of the hypotenuse enclosed between this leg and the height drawn from the top of the right angle.

Find MK and MN:

MK = √ (NK * KH) = √ (169 * 25) = √169 * √25 = 13 * 5 = 65 cm.

MN = √ (KN * HN) = √ (169 * 144) = 13 * 12 = 156 cm.

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