OH LORD THIS IS A NIGHTMARE FOR ME
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Answer:
104 m
Step-by-step explanation:
We can laboriously compute the positions of all of the midpoints, then find the difference of their coordinates, or we can see what those differences might be before we mess with any numbers.
The midpoint of JK is (J+K)/2; the midpoint of LM is (L+M)/2, so the difference in coordinates between these midpoints will be ...
(J+K-L-M)/2 = ((K-M) +(J-L))/2 . . . . path from JK to LM .. (P1)
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Similarly, the midpoint of KL is (K+L)/2, and the midpoint of MJ is (M+J)/2. Then the difference between these midpoints is ...
(K+L-M-J)/2 = ((K-M) -(J-L))/2 . . . . path from KL to MJ .. (P2)
So, the differences we want are ...
K-M = (1, 3) -(-1, -3) = (2, 6)
J-L = (-3, 1) -(5, -1) = (-8, 2)
And the differences in coordinates at the ends of the paths are ...
P1 = ((2, 6) +(-8, 2))/2 = (-3, 4)
P2 = ((2, 6) -(-8, 2))/2 = (5, 2)
The total path length is found by applying the Pythagorean theorem (distance formula) to the differences in coordinates ...
|P1| +|P2| = (10 m)×(√((-3)²+4²) +√(5² +2²)) = (10 m)(√25 +√29)
|P1| +|P2| ≈ (10 m)(10.385) ≈ 103.85 m ≈ 104 m
The total path length is about 104 meters.
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In the attached graph, the lengths of the cross-park paths in coordinate units are shown at lower left.
