Do the points joining L(6,4), M(-5,-3) and N(-6,8) from a
triangle ? Mention the type of triangle so formed .

Explanation

`L(6,4),M(-5,-3)` and `(-6,8)`
By distance formula
`LM=sqrt((-5-6)^(2)+(-3-4)^(2))`
`=sqrt((-11)^(2)+(-7)^(2))`
`=sqrt(121+49)`
`sqrt(170)`…………1
`MN=sqrt([-6-(-5)]^(2)+[8-(-3)]^(2))`
`=sqrt((-6+5)^(2)+(8+3)^(2))`
`=sqrt((-1)^(2)+(11)^(2))`
`=sqrt(1+121)`
`=sqrt(122)`......2
`LN=sqrt((-6-6)^(2)+(8-4)^(2))`
`=sqrt((-12)^(2)+4^(2))`
`=sqrt(144+16)`
`=sqrt(160)`........3
`MN+LN=sqrt(122)+sqrt(160)`
For a triangle to be formed the sum of two sides has to be greater than the third side.
We need to verify whether `sqrt(122)+sqrt(160)gtsqrt(170)`
Now
`(sqrt(122)+sqrt(160))^(2)=(sqrt(122))^(2)+2sqrt(122)xxsqrt(160)+(sqrt(160))^(2)`
`=122+2sqrt(122)xxsqrt(160)+160`
`=282+2sqrt(122)xxsqrt(160)`..........5
`(sqrt(170))^(2)=170`....5
`282gt170`
`:.282+2sqrt(122)xxsqrt(160)gt170`
`:.(sqrt(122)+sqrt(160))^(2)gt(sqrt(170))^(2)` .....[From 4 and 5]
`:.sqrt(122)+sqrt(160)gtsqrt(170)`
.....(Taking square roots on both the sides )
`:.DeltaLMN` is formed.
`:'LM~=MN~=LN`.
`:.DeltaLMN` is scalene.
The segment joining `L(6,4),M(-5,3)` and `N(-6,8)` forms a scalene triangle.