Hello, magentarita!

In a right triangle, where one of the legs equals 10,

the bisector of the opposite angle cuts the leg into the ratio of 2:3.

Find the area of the triangle.

This problem requires a seldom-used theorem about angle bisectors.

In a triangle, the angle bisector of an angle divides the opposite side

. . into segments proportional to the other two sides.

Code:

B
*
* * 2a
* *
2 * * D
* *
* * * 3a
* * *
* * *
A * * * * * * * C
3

Suppose we have with

If bisects , then

We have the following right triangle . . . Code:

B
o -
* * |
* * |
* 3 * |
3a * * 10
* o D |
* * * |
* * * |
* * 2 * |
* * * |
A o * * * * o -
2a C

We are given: .

bisects and divides in the ratio 2:3.

Hence,

Let:

Using Pythagorus, we have: .

Then: .

. . Hence: .

So we have: .

The area of the triangle is: .