If Betsy leaves Town A traveling due east for 6 miles to Town B, then turns due north and travels 9 miles to Town C, how far will her trip home be to Town A as she travels directly southwest?
If she travels directly southwest she will pass by Town A. It is not quite in that direction.
They probably just want the distance to Town A, which would be sqrt[(6^2) + (9)^2] = sqrt(117) = 10.8 miles
To find the distance of Betsy's trip home to Town A as she travels directly southwest, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse).
In this case, we can consider Betsy's trip home to Town A as a right-angled triangle. The 6-mile distance from Town A to Town B represents the horizontal side of the triangle (the base), and the 9-mile distance from Town B to Town C represents the vertical side of the triangle (the height). The distance from Town C back to Town A, which is the hypotenuse of the triangle, represents the shortest distance of Betsy's trip home.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
In this case, a = 6 miles and b = 9 miles.
c^2 = 6^2 + 9^2
c^2 = 36 + 81
c^2 = 117
To find the value of c, we take the square root of both sides of the equation:
c = sqrt(117)
c ≈ 10.82
Therefore, the distance of Betsy's trip home to Town A as she travels directly southwest is approximately 10.82 miles.