How do you show that a triangle with vertices $(13, - 2),(9, - 8),(5, - 2)$ is isosceles?

Hint: In the question given above, we have three vertices of a triangle and with the help of some formulas, we have to find whether the triangle is isosceles or not. In order to prove that a triangle is isosceles or not, we need to prove that two of its sides are equal, and for that we can use the formula to find the size of the sides with the help of the points in vertices.

Formula used: $ = \sqrt {{{({s_1} - {t_1})}^2} + {{({s_2} - {t_2})}^2}} $ where $({s_1},{s_2})\& ({t_1},{t_2})$ are the two vertices points of the sides.

Complete step-by-step solution:
One way of proving that it is an isosceles triangle is by calculating the length of each side since two sides of equal lengths means that it is an isosceles triangle.
Length of the side with vertices at\[(13, - 2)\& (9, - 8)\],
Putting values in the formula,
\[ \Rightarrow \sqrt {{{(13 - 9)}^2} + {{( - 2 + 8)}^2}} \]
Solving the brackets,
\[ \Rightarrow \sqrt {16 + 36} \]
Adding the numbers,
$ \Rightarrow \sqrt {52} $
Taking square roots,
$ \Rightarrow 2\sqrt {13} $ units
Length of the side with vertices at\[(9, - 8)\& (5, - 2)\],
Putting values in the formula,
\[ \Rightarrow \sqrt {{{(9 - 5)}^2} + {{( - 8 + 2)}^2}} \]
Solving the brackets,
\[ \Rightarrow \sqrt {16 + 36} \]
Adding the numbers,
$ \Rightarrow \sqrt {52} $
Taking square roots,
$ \Rightarrow 2\sqrt {13} $ units
Length of the side with vertices at \[(13, - 2)\& (5, - 2)\]
Putting values in the formula,
\[ \Rightarrow \sqrt {{{(13 - 5)}^2} + {{( - 2 + 2)}^2}} \]
Solving the brackets,
\[ \Rightarrow \sqrt {64} \]
Taking square roots,
$ \Rightarrow 8$ units
From the above calculations, you'll notice that length of \[(13, - 2)\& (9, - 8)\]and length of \[(9, - 8)\& (5, - 2)\] are the same.
Therefore, the triangle is isosceles.

Note: In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. The name derives from the Greek iso (same) and skelos (leg). The two equal sides are called the legs and the third side is called the base of the triangle. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.