Answer: short line: 30 miles, ling leg: 224 miles
Explanation:
Intro:
We will be using Pythagoras theorm, which states, if we have a 90 degree triangle which has 3 side as H, B and P defined as Hypotenuse, Base and Perpendicular, then H² = B² + P²
In your case, we know that hypotenuse is 226.
We also know that (A + B)² = A² + B² + 2AB
Given:
H = 226
B - P = 194
Solution:
B = 194 + P
So
(226)² = (194 + P)² + P²
51076 = (194)² + P² + (2 x 194 x P) + P²
51076 = 37636 + 2P² + 388P
51076 - 37636 = 2P² + 388P
13440 = 2P² + 388P
6720 = P² + 194P
P² + 194P - 6720 = 0
P² - 30P + 224P - 6720 = 0
P(P - 30) + 224(P - 30) = 0
(P - 30) x (P + 224) = 0
P = 30 or -224
As miles cannot be negative, we will choose answer as 30 miles.
Now we have H = 226 and P = 30
Using Pythagoras theorm:
(226)² = (30)² + B²
51076 = 900 + B²
50176 = B²
B = 224 miles.
So the missing miles are 224 miles and 30 miles