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The coordinates of the mid-points of the...

The coordinates of the mid-points of the sides of `DeltaPQR`, are `(3a,0,0), (0,3b,0)` and `(0,0,3c)` respectively, then the area of `DeltaPQR` is

A

`18sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`

B

`9sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`

C

`9/12sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`

D

`9/2sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`

Text Solution

Verified by Experts

The correct Answer is:
A

Let L,M,N bethe mid-points of the sides of `DeltaPQR`. Area of `DeltaLMN=1/2|vec(MN)xx vec(ML)|`
`=1/2|(-3bhati+3chatk) xx (3ahati-3bhatj)|`
`=1/2|9(bchati+cahatj+abhatk)|`
`=9/2sqrt((bc)^(2)+(ca)^(2)+(ab)^(2))`
Now, Area of `trianglePQR= 4 xx ("Area of" triangleLMN)`
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