Answer:
the perimeter is 10.99 units.
Step-by-step explanation:
We have a triangle with points in:
X // Y
N 2 // 0
P -1 // -2
M 4 // 0
We can solve for the side of the triangle by subtracting the points to calculate the distance between them, AKA the side of the triangle.
M - N = NM = 4 - 2 = 2 units of X
M - N = NM = 0 - 0 = 0 units of Y
This has a length of 2 units of X and 0 units of Y thus, we already have a line of size 2.
Next side:
N - P = 2 -(-1) = 3 units of x
N - P = 0 - (-2) = 2 units of Y
We have a length of 3 units of X and 2 of Y we use Pythagoras to get the length of NP:
![\sqrt{3^2+2^2} =NP\\\sqrt{9 + 4} = \sqrt{13}](https://tex.z-dn.net/?f=%5Csqrt%7B3%5E2%2B2%5E2%7D%20%3DNP%5C%5C%5Csqrt%7B9%20%2B%204%7D%20%3D%20%5Csqrt%7B13%7D)
Now, we do the same with MP:
M - P = 4 - (-1) = 5 units of X
M - P = 0 - (-2) = 2 units of Y
we solve for MP
![\sqrt{5^2+2^2} = \sqrt{25 + 4} = \sqrt{29}](https://tex.z-dn.net/?f=%5Csqrt%7B5%5E2%2B2%5E2%7D%20%3D%20%5Csqrt%7B25%20%2B%204%7D%20%3D%20%5Csqrt%7B29%7D)
We now add up:
![NM + NP + MP\\2 +\sqrt{13} + \sqrt{29} =\\10.99071608](https://tex.z-dn.net/?f=NM%20%2B%20NP%20%2B%20MP%5C%5C2%20%2B%5Csqrt%7B13%7D%20%2B%20%5Csqrt%7B29%7D%20%3D%5C%5C10.99071608)