The Quadrilateral is JKLM,
let
![M_{JK}, M_{KL}, M_{LM}, M_{JM},](https://tex.z-dn.net/?f=M_%7BJK%7D%2C%20M_%7BKL%7D%2C%20M_%7BLM%7D%2C%20M_%7BJM%7D%2C%20)
be the midpoints of JK, KL, LM and JM respectively.
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Given any 2 point P(m,n) and Q(k,l),<span>
the coordinates of the midpoint of the line
segment PQ are given by the formula:
, </span>
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thus the coordinates of points ![M_{JK}, M_{KL}, M_{LM}, M_{JM},](https://tex.z-dn.net/?f=M_%7BJK%7D%2C%20M_%7BKL%7D%2C%20M_%7BLM%7D%2C%20M_%7BJM%7D%2C%20)
are as follows:
![M_{JK}= (\frac{-3+1}{2}, \frac{1+3}{2})=(-1,2), \\\\M_{KL}= (\frac{1+5}{2}, \frac{3-1}{2}=(3,1), \\\\M_{LM}= (\frac{5-1}{2}, \frac{-1-3}{2})=(2, -2),\\\\ M_{JM}= (\frac{-3-1}{2}, \frac{1-3}{2})=(-2,-1)](https://tex.z-dn.net/?f=M_%7BJK%7D%3D%20%28%5Cfrac%7B-3%2B1%7D%7B2%7D%2C%20%5Cfrac%7B1%2B3%7D%7B2%7D%29%3D%28-1%2C2%29%2C%20%5C%5C%5C%5CM_%7BKL%7D%3D%20%28%5Cfrac%7B1%2B5%7D%7B2%7D%2C%20%5Cfrac%7B3-1%7D%7B2%7D%3D%283%2C1%29%2C%20%5C%5C%5C%5CM_%7BLM%7D%3D%20%28%5Cfrac%7B5-1%7D%7B2%7D%2C%20%5Cfrac%7B-1-3%7D%7B2%7D%29%3D%282%2C%20-2%29%2C%5C%5C%5C%5C%20M_%7BJM%7D%3D%20%28%5Cfrac%7B-3-1%7D%7B2%7D%2C%20%5Cfrac%7B1-3%7D%7B2%7D%29%3D%28-2%2C-1%29)
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The distance between any 2 points P(a,b) and
Q(c,d) in the coordinate plane, is given by the formula:<span>
</span>
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thus the distances connecting the opposite entrances can be calculated as follows:
![|M_{JK},M_{LM}|= \sqrt{ (-1-2)^{2} + (2-(-2))^{2} }= \sqrt{9+16}=5](https://tex.z-dn.net/?f=%7CM_%7BJK%7D%2CM_%7BLM%7D%7C%3D%20%5Csqrt%7B%20%28-1-2%29%5E%7B2%7D%20%2B%20%282-%28-2%29%29%5E%7B2%7D%20%7D%3D%20%5Csqrt%7B9%2B16%7D%3D5%20%20)
![|M_{KL}M_{JM}|= \sqrt{ (3-(-2))^{2} + (1-(-1))^{2}}= \sqrt{25+4}= \sqrt{29}=5.39](https://tex.z-dn.net/?f=%7CM_%7BKL%7DM_%7BJM%7D%7C%3D%20%5Csqrt%7B%20%283-%28-2%29%29%5E%7B2%7D%20%2B%20%281-%28-1%29%29%5E%7B2%7D%7D%3D%20%5Csqrt%7B25%2B4%7D%3D%20%5Csqrt%7B29%7D%3D5.39%20)
Thus the total distance of the paths joining the opposite entrances is
5+5.39 units = 50 m + 53.9 m = 104 m (rounded to the nearest meter)
Answer: 104 m