In a right-angled triangle ABC, the height CH, drawn from the vertex of the right angle C

| March 10, 2021 | education

In a right-angled triangle ABC, the height CH, drawn from the vertex of the right angle C, divides the hypotenuse into two segments AH = 16 cm and BH = 25 cm, find CH, AC, AC. Area ACH, area BCH.

Let us apply for the solution the property of the height of a right-angled triangle drawn to the hypotenuse. The square of the height is equal to the product of the lengths of the segments by which it divides the hypotenuse.

CH ^ 2 = AH * BH = 16 * 25 = 40.

CH = 20 cm.

In right-angled triangles АСН and ВСН, the side of СН is common, and since АН is more than ВН, the smaller leg of the triangle ABC is a segment BC

In a right-angled triangle ACН, according to the Pythagorean theorem, we determine the length of the hypotenuse AC.

AC ^ 2 = CH ^ 2 + AH ^ 2 = 20 ^ 2 + 16 ^ 2 = 400 + 256 = 656.

AC = 4 * √41 cm.

In a right-angled triangle ВСН, according to the Pythagorean theorem, we determine the length of the hypotenuse ВС.

BC ^ 2 = CH ^ 2 + BH ^ 2 = 20 ^ 2 + 25 ^ 2 = 400 + 625 = 1025.

BC = 5 * √41 cm.

Ssn = AН * CH / 2 = 16 * 20/2 = 160 cm2.

Svsn = ВН * CH / 2 = 16 * 25/2 = 200 cm2.

Answer: The height of the triangle is 20 cm, 4 * √41 cm, Ssn = 160 cm2, Ssn = 200 cm2.



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