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  • please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22.4 question 6 maths textbook solution

please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22.4 question 6 maths textbook solution

Answers (1)

The internal bisectors of the angles of a triangle are concurrent.

Hint:

With the help of vector algebra.

Given:

Concurrent bisector denotes the quality of internal bisectors.

 

Let ABC be the triangle and \vec{\alpha},\vec{\beta },\vec{\gamma } be the position vectors of the vertices A,B and C respectively. Let AD,BE and CF be the internal bisectors of \angle A,\angle B and\angle C respectively.

We know    that D divides BC in the ratio of AB:AC that is c:b.

Then,

Postiton vector of D is  \frac{c\vec{\gamma }+b\vec{\beta }}{c+b}

Position vector of E is \frac{c\vec{\gamma }+b\vec{\beta }}{c+b}   and of F is \frac{a\vec{\alpha }+b\vec{\beta }}{a+b}

The point dividing AD in the ratio  b+c : a  is \frac{a\vec{\alpha }+b\vec{\beta }+c\vec{\gamma }}{a+b+c}

The point dividing BE in the ratio  a+c : b  is \frac{a\vec{\alpha }+b\vec{\beta }+c\vec{\gamma }}{a+b+c}

The point dividing CF in the ratio  a+b : c  is \frac{a\vec{\alpha }+b\vec{\beta }+c\vec{\gamma }}{a+b+c}

Since the point \frac{a\vec{\alpha }+b\vec{\beta }+c\vec{\gamma }}{a+b+c} lies on all the three internal bisectors AD,BE and CF .Hence the internal bisectors are concurrent.

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