Shorties distance between two points

Question

John lives 3 miles north and 2 miles east of school. Jim lives 5 miles south and 4 miles west of school. What is the shortest distance between their houses?

Answer 1

This problem is pretty clear when you put it on an x-y graph.

 

3 miles north and 2 miles east    translates to the point     x = 2  y = 3  
5 miles south and 4 miles west   translates to the point     x = -4  y = -5

The shortest distance between their two houses is the straight line segment between (2,3)and (-4,-5).

Now, each of the points was plotted as a right triangle, so the Pythagorean theorem may be used to find the third side of the two triangles (they add up to the total diatance).

  D1^2 = 2^2 + 3+2
  D2^2 = 4^2 + 5^2
   D  =  D1  + D2  =  SQRT(2^2 + 3^2)   + SQRT ( 4^2 + 5^2)

Now you could calculate that, or you could realize that the two points themselves are on a big triangle with sides of (2 – (-4)) = 6   and  (3 – (-5)) = 8  and find the hypotenuse of that big triangle.

(p.s., most math problems at this level have “magic numbers” that make the calculations appropriately easy to do, so you can concentrate on the solution, not the calculations)

Now    D  = SQRT( 6^2 + 8^2)
             D = SQRT ( 100)
            D = 10                               Whew! That’s  much easier to calculate!