# The bisector of a right-angled triangle divides the hypotenuse into 20 cm and 15 cm segments. Find the area of the triangle.

September 14, 2021 | education

| Since CM is the bisector of an angle, then by the property of the bisector of a triangle: AC / AM = BC / BM.

AC / 15 = BC / 20.

AC / 3 = BC / 4.

AC / BC = 3/4.

Let the length of the AC segment be 3 * X cm, then the length of the BC segment = 4 * X cm.

The length of the hypotenuse AB = AM + BM = 15 + 20 = 35 cm.

In a right-angled triangle ABC, according to the Pythagorean theorem:

AB ^ 2 = AC ^ 2 + BC ^ 2.

1225 = 9 * X ^ 2 + 16 * X ^ 2.

25 * X ^ 2 = 1225.

X ^ 2 = 1225/25 = 49.

X = 7 cm.

AC = 3 * 7 = 21 cm.

BC = 4 * 7 = 28 cm.

Then the area of the triangle is equal to: Sавс = АС * ВС / 2 = 21 * 28/2 = 294 cm2.

Answer: The area of triangle ABC is 294 cm2.

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