Question

How do I find the perimeter of a triangle with vertices \[\left( { - 2,2} \right),\left( {0,4} \right),\left( {1 - ,4} \right)\]?

Answer 1

Hint: In the given question, we have been given that there is a triangle whose coordinates of the vertices are given. We have to find the perimeter of the triangle. To find the perimeter of the given triangle, we are going to first find the length of the sides using the distance formula. Then when we have calculated the length of all sides, we are simply going to add them all up and we are going to have our answer.

Complete step by step answer:
We are going to use the distance formula:
Let there be two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\]. Then the distance between the points is:
\[{d_i} = \sqrt {{{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{x_2} - {x_1}} \right)}^2}} \]

Let the sides being represented by the vertices be \[A\left( { - 2,2} \right),B\left( {0,4} \right),C\left( {1, - 4} \right)\].
First, let us calculate the length of the sides.
\[AB = \sqrt {{{\left( {0 + 2} \right)}^2} + {{\left( {4 - 2} \right)}^2}} = \sqrt {4 + 4} = 2\sqrt 2 \]
\[BC = \sqrt {{{\left( {1 - 0} \right)}^2} + {{\left( { - 4 - 4} \right)}^2}} = \sqrt {65} \]
\[CA = \sqrt {{{\left( {1 + 2} \right)}^2} + {{\left( { - 4 - 2} \right)}^2}} = \sqrt {9 + 36} = 3\sqrt 5 \]
Now, perimeter means sum of all sides.
Hence, \[P = 2\sqrt 2 + \sqrt {65} + 3\sqrt 5 \] units

Note: In the given question, we had to find the perimeter of a triangle whose coordinates were given. The point where the things may go wrong is if we do not pay attention to the sign of the value of the coordinates. To the beginners it might seem that it is not such a big issue, but it totally flips the answer if we make the slightest mistake of a wrong sign. We pay attention to that, write the formula correctly, and follow the procedure, we are good to go.