The longer route is 4 miles further than the shorter route.
What is meant by the Pythagorean theorem?
According to the concept, the square of the hypotenuse of a right triangle equaled the square of the other sides. The formula is
[tex]c^{2} = a^{2} + b^{2}[/tex],
where c is the hypotenuse length and a and b are the other two sides' lengths.
Calculate the difference between the longer route and the shorter route:
By using the Pythagorean theorem,[tex]c^{2} =a^{2} +b^{2}[/tex]
Your hypotenuse is the straight path from Crimson to Sunshine, and your legs are the roadways to and from Happy Ville. [tex]c^{2} =6^{2} +8^{2}[/tex]
Take the square root of both sides of the equation to eliminate the exponent on the left side: [tex]c=\pm \sqrt{6^{2}+8^{2}}[/tex]
Raise 6 to the power of 2:[tex]c=\pm \sqrt{36+8^{2}}[/tex]
Raise 8 to the power of 2: [tex]c=\pm \sqrt{36+64}[/tex]
Add 36 and 64: [tex]c=\pm \sqrt{100}[/tex]
Rewrite 100 as [tex]10^{2}[/tex]
[tex]c=\pm \sqrt{10^{2}}[/tex]
Pull terms out from under the radical, assuming positive real numbers.
[tex]c=\pm 10[/tex]
So the distance between Crimson and Sunshine is 10 miles.It asks for the shortest path, either to Happy ville and then to Sunshine (through the legs) or to Sunshine immediately (hypotenuse),
Legs = 6+8 = 14
Hypotenuse = 10
So getting straight to Sunshine would be faster than going to Happyville than Sunshine because going straight to Sunshine is 10 miles against 14 miles for going to Happyville than sunshine.The longer path was 14 miles, while the shorter way was 10 miles, so subtract the two.14−10 = 4Hence, the longer route is 4 miles further than the shorter route.
Learn more about the Pythagorean theorem here:brainly.com/question/20545047
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