# The height drawn to the hypotenuse of a right-angled triangle is 5 cm, and the length of one

The height drawn to the hypotenuse of a right-angled triangle is 5 cm, and the length of one of the legs is 13 cm. Calculate the length of the hypotenuse of the triangle.

The height of the CH forms two right-angled triangles АСН and ВСН.

From the right-angled triangle ACH, according to the Pythagorean theorem, we determine the length of the leg AH.

AH ^ 2 = AC ^ 2 – CH ^ 2 = 169 – 25 = 144.

AH = 12 cm.

Let us prove the similarity of triangles ACH and BCH.

Let the value of the angle HAC of the triangle ABC be equal to X0, then the angle ACH = (90 – X) 0.

Angle АСВ = 90, then angle ВСН = (90 – (90 – X) = X0.

The acute angles of the right-angled triangles АСН and ВСН are equal, then the triangles are similar in acute angle.

Then in similar triangles АН / СН = СН / ВН.

CH ^ 2 = AH * BH.

BH = CH ^ 2 / AH = 25/12 = 2 (1/12) cm.

Then AB = AH + BH = 12 + 2 (1/12) = 169/12 = 14 (1/12) cm.

Answer: The length of the hypotenuse AB is 14 (1/12) cm.

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