Question

Show that the triangle with vertices $(5,4,-1)$, $(3,6,-1)$, and $(3,4,1)$ is equilateral.

Short answer

The triangle with given vertices is equilateral.

Step-by-step solution

Step 1. Given information

Given vertices of the triangle are $(5,4,-1)$, $(3,6,-1)$, and $(3,4,1)$.

Let A= $(5,4,-1)$, B= $(3,6,-1)$, and C=$(3,4,1)$

Step 2. Calculating the Distance between AB, BC and CD

Distance AB:

A= (5, 4, −1), B= (3, 6, −1)

AB = $\sqrt{{(3-5)}^2+{\left(6-4\right)}^2+{\left(-1+1\right)}^2}$

=$\sqrt{4+4+0}$

= $\sqrt{8}$ units

Distance BC:

B= (3, 6, −1) ,C=(3, 4, 1)

BC =$\sqrt{{(3-3)}^2+{\left(4-6\right)}^2+{\left(1+1\right)}^2}$

= $\sqrt{0+4+4}$

= $\sqrt{8}$ units

Distance AC:

A= (5, 4, −1) ,C=(3, 4, 1)

AC = $\sqrt{{(3-5)}^2+{\left(4-4\right)}^2+{\left(1+1\right)}^2}$

=$\sqrt{4+0+4}$

= $\sqrt{8}$ units

Here the sides of triangle ABC are equal in length.

Therefore, the given triangle is an equilateral triangle.