Trig question

Question

Instead of walking along two sides of a rectangular field, Fran took a shortcut along the diagonal, thus saving distance equal to half the length of the longer side. Find the length of the long side of the field, given that the the length of the short side is 156 meters.

 

 

I don't know how to start this problem. Help would be appreciated

Answer 1

I think Michael misread the problem.

It's going to be a little more messy!

 

The shorter path saves half the length of the longer side.

X = the diagonal walked

L = The longer side

156 = the shorter side.

The long route is L + 156.

Subtracting the shortcut distance, we get an expression for the distance saved.

L + 156 - X

That is half the length of the longer side, so:

L + 156 - X = L/2

Using the Pythagorean theorem, we get an expression for the diagonal X:

X = √(L²+156²)

Substitute this for X above.

L + 156 - √(L²+156²) = L/2

Isolate the radical on one side, then square both sides.

L - L/2 + 156 = √(L²+156²)

(L/2 + 156)² = L² + 156²

L²/4 +156L + 156² = L² +156²

0 = L² - L²/4 - 156L

3L²/4 - 156L = 0      (I swapped sides, too.)

L(3L/4 - 156) = 0

L = 0       Doesn't make sense.  Ignore it.

3L/4 - 156 = 0

L = 4/3 (156) = 208

 

Check:

X = √(156² + 208²) = 260      shortcut

156 + L = 156 + 208 = 364    long route

364 - X = 364 -260 = 104

The savings is half of L.

 

 

Answer 2

The diagonal that is created by connecting the two sides together results in a right triangle.  We can use the Pythagorean theorem.

 

(shorter side)² + (longer side)² = (distance walked)²

 

Let distance walked = x

Let longer side = 2x

shorter side = 156

 

Using these values, you have the equation

 

(2x)² + 156² = x²

 

4x² + 156² = x²

 

 

Finish solving the equation for x.  Then once you have your value of x, multiply it by 2.