The height of a right-angled triangle, drawn to the hypotenuse, divides it into segments 18 cm and 32 cm.

| February 22, 2021 | education

The height of a right-angled triangle, drawn to the hypotenuse, divides it into segments 18 cm and 32 cm. Find the length of the segments into which the bisector of the larger acute angle of the triangle divides this height.

From the property of the height of a right-angled triangle drawn from the apex of an obtuse angle, we determine the height of the VN.

BH ^ 2 = AH * CH = 32 * 18 = 576.

BH = 24 cm.

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

BC ^ 2 = BH ^ 2 + CH ^ 2 = 24 ^ 2 + 18 ^ 2 = 576 + 324 = 900.

BC = 30 cm.

Since SC is the bisector of the angle C of the triangle ABC, it is also the bisector of the angle C of the triangle BCH. According to the property of the angle bisector, the segments into which it divides the side are proportional to the adjacent sides.

Let the segment OH = X cm, then the segment BO = 24 – X cm.

CH / OH = BC / OB.

18 / X = 30 / (24 – X).

30 * X = 432 – 18 * X.

48 * X = 432.

X = OH = 432/48 = 9 cm.

VO = 24 – 9 = 15 cm.

Answer: The bisector divides the height into 9 cm and 15 cm segments.



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