Q. 51 - Calculus

Question

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

Step-by-Step Solution

Answer

The triangle with given vertices is equilateral.

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.