Question

Wanda is trying to locate the Fermat point P of triangle ABC, where A is at the origin, B is at (8,-1), and C is at (5,4) (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at P = (4,2), and computes the sum of the distances from P to the vertices of $\triangle ABC$. If she obtains m + n\sqrt{5},
where m and n are integers, what is m + n?

Answer 1

9514 1404 393

Answer:

  8

Step-by-step explanation:

The distance formula is used to find the distances to each of the points:

  d = √((x2-x1)² +(y2-y1)²)

  d = √((4-0)² +(2-0)²) +√((4-8)² +(2-(-1))²) +√((4-5)² +(2-4)²)

  d = √20 +√25 +√5 = 2√5 +5 +√5

  d = 5 +3√5

Comparing to the form ...

  d = m +n√5

we see that m=5, n=3. The the sum m+n = 5+3 = 8.