Question

Show that the triangle with vertices (1, 2, −2), (−3, 2, −6), and (−3, 6, −2) 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 (1, 2, −2), (−3, 2, −6), and (−3, 6, −2)

Let A=(1, 2, −2), B=(−3, 2, −6), and C=(−3, 6, −2)

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

Distance AB:

A=(1, 2, −2), B=(−3, 2, −6),

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

=\(\sqrt 32\) units

Distance BC:

B= (−3, 2, −6), C=(−3, 6, −2)

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

=\(\sqrt 32\) units

Distance AC:

A= (1, 2, −2), C=(−3, 6, −2)

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

= \(\sqrt 32\) units

Here the sides of triangle ABC are equal in length.

Therefore, the given triangle is an equilateral triangle.