The perimeter of triangle AMNP is 54 + 5x units.
The perimeter of a triangle is the total length of all its sides. To find the perimeter of triangle AMNP, we need to find the lengths of all three sides.
We are given that the length of side NP is 44 units. We are also given that side MN is equal to 2(RS), where R is the midpoint of segment NP and S is the midpoint of segment MP.
We can see from the diagram that triangle NPR is a right triangle, with NP as the hypotenuse and NR and PR as the legs. We are given that the length of PR is x + 4 units. Using the Pythagorean theorem, we can find the length of NR:
NR^2 = NP^2 - PR^2
NR^2 = 44^2 - (x + 4)^2
NR^2 = 1936 - (x^2 + 8x + 16)
NR^2 = 1920 - x^2 - 8x
We are also given that MS = 5x - 34. Since RS is half of NP, we know that RS = NP/2 = 44/2 = 22.
Now we can find the length of MN:
MN = 2(RS) = 2 * 22 = 44
Finally, we can find the perimeter of triangle AMNP:
Perimeter = MN + NP + MP
Perimeter = 44 + 44 + (5x - 34)
Perimeter = 88 + 5x - 34
Perimeter = 54 + 5x
Therefore, the perimeter of triangle AMNP is 54 + 5x units.