Question

If the points $(0,7,10),(-1,6,6)and(-4,9,6)$ are the vertices of a triangle, then the triangle is _____

• A. Right angled
• B. Isosceles right angled
• C. Equilateral
• D. None of these

Answer 1

The correct option is B

Isosceles right angled

Let A $\equiv (0,7,10),B\equiv (-1,6,6),C\equiv (-4,9,6)$

Distance between two points $(x_1,y_1,z_1)$ $\&$ Q $(x_2,y_2,z_2)$ is

PQ= $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

so,AB=$\sqrt{(0+1{)}^2+(7-6{)}^2+(10-6{)}^2}$ = $\sqrt{18}$ = 3$\sqrt{2}$

similarly,$BC\equiv 3\sqrt{2}\&AC=6$

clearly,we see that $A{B}^2+B{C}^2=A{C}^2$

$(3\sqrt{2}{)}^2+(3\sqrt{2}{)}^2=36$

$18+18=36$

$36=36$

$\therefore \angle ABC={90}^{\circ }$

Triangle formed by these coordinates is right angle triangle.

Also AB=BC

Hence,$△ABC$ is right angled isosceles triangle

option B is correct