To find the distance between Melissa and Caroline, we must first form the other two sides of the triangle:
We are given the information that Melissa is 3 blocks west and 4 blocks south of Dawn. Form a triangle. We know that the hypotenuse has a length of 5 because of the 3-4-5 Pythagorean Triple (or you could use SOH-CAH-TOA, but knowing the PTs are really useful in problems like these).
We can now use the sine function to determine the angles. The only angle we need for this problem is the one formed by the side length of 3 and the hypotenuse. arcsin(4/5) is approximately 53.13. If we subtract this from 180, we will get the largest angle of the triangle formed by the distances between all three friends: 126.87 degrees.
Now that we have one side and one angle, we need the distance between Dawn and Caroline so we can use the Law of Cosine:
We are told that Caroline is 5 blocks east and 1 block north of Dawn. Form a triangle. By using the Pythagorean Theorem (5²+1²=c², c=√26), we know that the hypotenuse of this specific triangle is √26. This hypotenuse is the final side we need for the largest triangle (the focus of this problem)
Now, we can use the Law of Cosine:
c²=a²+b²-2·a·b·cosC
c²=5²+√26²-2·5·√26·cos(126.87)
c²=51-(-30.59)
c²=81.59
c≈ 9.03 blocks
