Are the points `A(3,6,9),\ B(10 , 20 , 30)a n d\ C(25 ,-41 ,5),` the vertices of a right angled triangle?

Explanation

Given that the points are `A(3,6,9),\ B(10 , 20 , 30) a n d\ C(25 ,-41 ,5),`
We have to show that these points are vertices of a right angled triangle.
`AB=sqrt((3-10)^2+(6-20)^2+(9-30)^2)`
`=sqrt(49+196+441)`
`=sqrt(686)`
`BC=sqrt((10-25)^2+(20+41)^2+(30-5)^2)`
`=sqrt(225+3721+625)`
`=sqrt(4571)`
`CA=sqrt((25-3)^2+(-41-6)^2+(5-9)^2)`
`=sqrt(484+2201+16)`
`=sqrt(2701)`
Hence `BC^2=AB^2+CA^2`
therefore `A,B,C ` are the vertices of a right angled triangle.