# The height of the right triangle is 10cm. drawn to the hypotenuse is 4 cm, and divides it into segments

The height of the right triangle is 10cm. drawn to the hypotenuse is 4 cm, and divides it into segments, the difference of which is 6 cm. Find the sides of the triangle.

Let us prove the similarity of the triangles AСН and BCH.

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

Angle АСВ = 900, then angle ВСН = (90 – (90 – X) = X0.

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

Then AH / CH = CH / BН.

AH * BH = CH * CH = 16 cm.

Let the length of the segment BH = X cm, then AH = (X + 6) cm.

Then (X + 6) * X = 16.

X2 + 6 * X – 16 = 0.

X = BH = 2 cm, then AH = 2 + 6 = 8 cm.

AB = 8 + 2 = 10 cm.

In a right-angled triangle ACН, AC ^ 2 = AH ^ 2 + CH ^ 2 = 64 + 16 = 80.

AC = √80 = 4 * √5 cm.

In a right-angled triangle BCH, BC ^ 2 = BH ^ 2 + CH ^ 2 = 4 + 16 = 20.

AC = √20 = 2 * √5 cm.

Answer: The sides of the triangle are 4 * √5 cm, 2 * √5 cm, 10 cm.

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