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.
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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.
Hello, magentarita!
Quote:
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.
Suppose we haveCode:B
*
* * 2a
* *
2 * * D
* *
* * * 3a
* * *
* * *
A * * * * * * * C
3
with
Ifbisects
, then
We have the following right triangle . . .Code:B
o -
* * |
* * |
* 3 * |
3a * * 10
* o D |
* * * |
* * * |
* * 2 * |
* * * |
A o * * * * o -
2a C
We are given: .
in the ratio 2:3.
Hence,
Let:
Using Pythagorus, we have: .
Then: .
. . Hence: .
So we have: .
The area of the triangle is: .
Just out of curiousity , is there a name for this theorem and is there a proof for this theorem . THanks .
Wonderful job as usual. Thanks.Quote:
Originally Posted by Soroban
Hello, magentarita!
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.
Suppose we haveCode:B
*
* * 2a
* *
2 * * D
* *
* * * 3a
* * *
* * *
A * * * * * * * C
3
If
We have the following right triangle . . .We are given: .Code:B
o -
* * |
* * |
* 3 * |
3a * * 10
* o D |
* * * |
* * * |
* * 2 * |
* * * |
A o * * * * o -
2a C
Hence,
Let:
Using Pythagorus, we have: .
Then: .
. . Hence: .
So we have: .
The area of the triangle is: .