What are the lengths of the two legs of the sidewalk

Question

An L-shaped sidewalk from the library to the gym on a high school campus is 90 yards long, as shown in the figure. By cutting diagonally across the grass, students shorten the walking distance to 70 yards. What are the lengths of the two legs of the sidewalk? (Round your answers to one decimal place.)

Answer 1

We're essentially looking at a right triangle here. They gave us the hypotenuse as 70ft and told us that the long way around is 90 ft. We can think of the long way as being the sum of the two legs of the triangle. In other words a + b = 90 and c = 70.

Plugging these numbers into the pythagorean theorem, we get a^2+b^2=70^2=4900. Remeber that a+b=90, which we can rewrite as a=90-b. Using substitution, we get (90-b)^2+b^2=8100-180b+2b^2=4900. Moving all the terms to one side and dividing everything by 2 (for simplicity) gives us the quadratic equation b^2-90b+1600=0. Using the quadratic formula, we get b=(90±√(8100-6400))/2=(90±√(1700))/2=45±5√17.

Notice that (45+5√17)+(45-5√17)=90. We weren't given any information on which of the two legs is longer so all we can say is that one leg has length 45+5√17 and the other has length 45-5√17.