Question

The points $A(-4,9,6),B(-1,6,6)$ and $C(0,7,10)$ forms which type of triangle?

Answer 1

Find the type of triangle formed.

The given points are:

$A(-4,9,6),B(-1,6,6)$ and $C(0,7,10)$

Now,

$$\begin{gathered} \overrightarrow{AB}=\left(-1+4\right)i+\left(6-9\right)j+\left(6-6\right)k \\ =3i-3j \\ \Rightarrow {\left|AB\right|}^2=3^2+3^2 \\ =9+9 \\ =18 \\ \overrightarrow{Bc}=\left(0+1\right)i+\left(7-6\right)j+\left(10-6\right)k \\ =i-j+4k \\ \Rightarrow {\left|\overrightarrow{BC}\right|}^2=1^2+1^2+4^2 \\ =18 \\ \overrightarrow{AC}=\left(0+4\right)i+\left(7-9\right)j+\left(10-6\right)k \\ =4i-2j+4k \\ \Rightarrow {\left|\overrightarrow{AC}\right|}^2=4^2+2^2+4^2 \\ =16+4+16 \\ =36 \end{gathered}$$

Since, $A{B}^2+B{C}^2=A{C}^2,\left|\overrightarrow{BC}\right|=\left|\overrightarrow{AB}\right|$

Hence, the triangle formed from the above-given points is a right-angled, isosceles triangle.