# The bisector of the right angle of a right-angled triangle divides the hypotenuse

**The bisector of the right angle of a right-angled triangle divides the hypotenuse in a ratio of 3: 4. Calculate the area of the triangle if the length of the hypotenuse is 35cm.**

Let the length of the segment AM = 3 * X cm, then, by condition, the length of the segment BM = 4 * X cm.

AM + BM = AB = 35 cm.

3 * X + 4 * X = 35.

7 * X = 35.

X = 35/7 = 5.

Then AM = 3 * 5 = 15 cm, BM = 4 * 5 = 20 cm.

By the property of the bisector of a triangle:

AC / AM = BC / VM.

AC / 15 = BC / 20.

AC / BC = 3/4.

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

Then, by the Pythagorean theorem, AB ^ 2 = AC ^ 2 + BC ^ 2.

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

Y ^ 2 = 1225/25 = 49.

Y = 7.

Then AC = 3 * 7 = 21 cm, BC = 4 * 7 = 28 cm.

Savs = AC * BC / 2 = 21 * 28/2 = 294 cm2.

Answer: The area of the triangle is 294 cm2.