Question

You and your friend start walking from the same point (3, 3), shown in the graph. Your friend walks 5 miles east and then 2 miles due south. You walk 6 miles west and then 1 mile due north.Who is directly farther from the starting point?ANeither, you are both 7 miles away from the starting point.BNeither, you are both √7 miles away from the starting point.CYour friend is farther away, √29 miles.DYou are farther away, √37 miles.

Answer 1

If the friend walked 5 miles east and then 2 miles south, his position in the coordinate plane is (3+5, 3-2) = (8, 1).

If you walked 6 miles west and 1 mile north, your position in the coordinate plane is (3-6, 3+1) = (-3, 4).

Let's calculate the distance from each point to the starting point, using the formula below for the distance between two points (x1, y1) and (x2, y2):

$d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}$

For points (3, 3) and (8, 1), we have:

$\begin{gathered} d_1=\sqrt[]{(8-3)^2+(1-3)^2} \\ d_1=\sqrt[]{5^2+(-2)^2} \\ d_1_{}=\sqrt[]{25+4}=\sqrt[]{29} \end{gathered}$

Now, for points (3, 3) and (-3, 4), we have:

$\begin{gathered} d_2=\sqrt[]{(-3-3)^2+(4-3)^2} \\ d_2=\sqrt[]{(-6)^2+1^2} \\ d_2=\sqrt[]{36+1}=\sqrt[]{37} \end{gathered}$

The distance 2 (your distance) is greater than the distance 1 (friend's distance).

Therefore you are directly farther from the starting point, with a distance of √37 miles (Option D).