Find the distance between the points \[\left( {3,7} \right)\] and \[\left( { - 2, - 5} \right)\].

Hint:In the given question, we have been given the coordinates of two points. There is a line joining the two points. We have to find the distance between the given two points. We can use the distance formula to find the distance between two points which is\[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]. The distance between the two points is always positive.

Complete step by step answer:
Segments which have the same length are called congruent segments. We can easily calculate the distance between two points. The points can be considered as \[{x_1}\],\[{x_2}\], \[{y_1}\]and \[{y_2}\].We can substitute the actual value of points into the distance formula given as,
\[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Substituting the values we have,
\[\sqrt {{{\left( {2 - \left( { - 3} \right)} \right)}^2} + {{\left( {\left( { - 5} \right) - 7} \right)}^2}} \]
Multiplying the two subtractions we get a positive hence we have,
\[ \Rightarrow \sqrt {{{\left( {2 + 3} \right)}^2} + {{\left( {\left( { - 5} \right) - 7} \right)}^2}} \]

Now on adding we obtain,
\[ \Rightarrow \sqrt {{5^2} + {{\left( {\left( { - 5} \right) - 7} \right)}^2}} \]
Now solving the square and writing the equation gives us the following,
\[ \Rightarrow \sqrt {25 + {{\left( {\left( { - 5} \right) - 7} \right)}^2}} \]
Going to the second part and solving the same way we can have,
\[\sqrt {25 + {{\left( { - 12} \right)}^2}} \\
\Rightarrow \sqrt {25 + 144} \\
\Rightarrow \sqrt {169} \\
\Rightarrow \sqrt {{{13}^2}} \\
\therefore 13 \\ \]
Hence, the distance between the points \[\left( {3,7} \right)\] and \[\left( { - 2, - 5} \right)\] is \[13\].

Note: The formula for calculating the distance between the two points is important. The linear distance between the two points is the square root of the sum of the squared values of the “x” axis distance and the “y” axis distance. The Distance Formula is a variant of the Pythagorean Theorem that we used in geometry. Segments which have the same length are called congruent segments. We can easily calculate the distance between two points. Substitution in formula should be done right.