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

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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