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If p , q , r are the legths of the inter...

If p , q , r are the legths of the internal bisectors of angles A ,B , C respectively of a `!ABC` , then area of ABC

A

`1/a+1/b-1/c`

B

`1/a+1/c-1/b`

C

`1/a+1/b+1/c`

D

`1/b+1/c-1/a`

Text Solution

Verified by Experts

We have,
`Delta=`Area of `Delta` ABD + Area of `Delta` ACD
`rArrDelta=1/2pcsinA/2+1/2pbsinA/2`
`rArr1/2bcsinA=1/2pcsinA/2+1/2pbsinA/2`
`rArr1/2bc(2sinA/2cosA/2)=1/2pcsinA/2+1/2pbsinA/2`
`rArr2bccosA/2=pc+pbrArr2/pcosA/2=1/b+1/c`
Similarly, we have
`2/qcosB/2=1/a+1/c` and `2rcosC/2=1/a+1/b`
`therefore2/pcosA/2+2/qcosB/2+2/rcosC/2=(1/b+1/c)+(1/a+1/c)+(1/a+1/b)`
`rArr1/pcosA/2+1/qcosB/2+1/rcosC/2=1/a+1/b+1/c`
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