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The centroid of a triangle ABC is at the...

The centroid of a triangle ABC is at the point `(1,1,1)`. If the coordinates of A and B are `(3,-5,7)` and `(-1,7,-6)`, respectively, find the coordinates of the point C.

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The correct Answer is:
`(15,32)`or `(20,7)`
Let the coordinates of point `A be (h,k)`. Midpoint of BC is `D(35//2,39//2)`.
Slope of AD is
`((39)/(2)-k)/((35)/(2)-h)=-5`
`rArr 39-2k=-5(35-3h)`
`rArr39-2k=-175+10h`
`rArr5h+k=107`
Also, area of `DeltaABC` is 70 sq.units.
`therefore|{:(h,,h,,1),(12,,19,,1),(23,,20,,1):}|=+-140`
`rArr11k-h=337` (1)
or `11k-h=57` (2)
Solving (1) and (2), we get `h=15,k=32 `
Solving (1) and (3), we get `h=20,k=7`
So, possible coordinates of A are `(15,32) or (20,7)`.
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