Respuesta :

The Pythagorean theorem is: a²+b²=c², and "c" will always be the hypotenuse or the longest side of the triangle.

In order to find MN you can make a triangle, and then you can find the lengths of the other sides(a and b).

You should get a = 4 and b = 6. Next you plug these numbers into the equation.

(4)² + (6)² = c²

16 + 36 = c²

52 = c²

√52 = c

Ver imagen itmye84

The length of line segment [tex]\overline{MN}[/tex] is  [tex]\sqrt{52}[/tex]  units.

Given in graph:

  • Starting point of [tex]\overline{MN}[/tex] :  (-1,-2)
  • End point of [tex]\overline{MN}[/tex] :  (5,2)
  • To find: The length of the line segment [tex]\overline{MN}[/tex] .

Calculation of length of given line segment:

The length of [tex]\overline{MN}[/tex]  = distance between its endpoints.

The formula for distance between (x,y) and (a,b) is given as:

[tex]Distance (\:(x,y), (a,b)\:) = \sqrt{(x - a)^2 + (y-b)^2}[/tex]

Thus, applying this formula on endpoints of [tex]\overline{MN}[/tex] , we get length of [tex]\overline{MN}[/tex] as:

[tex]\begin{aligned}\text{Length of line segment MN } &= Distance( \: (-1,-2), (5,2) \: )\\&= \sqrt{(-1-5)^2 + (-2 -2)^2}\\&= \sqrt{6^2 + 4^2}\\&= \sqrt{52}\\\end{aligned}[/tex]

Thus, the length of line segment [tex]\overline{MN}[/tex] is [tex]\sqrt{52}[/tex]  units.

Learn more about lengths of line segments here:

https://brainly.com/question/1696052