Hi, I don't remember how to solve this. Please help me remember.

"I live 3 miles north and 1 mile west of the mall. Ellen lives 2 miles south and 5 miles east of the mall. What is the shortest distance between our houses?"

Thanks in advance,
Haley

Think about it as a graph and the directions as points. So the center (0,0) in this problem is represented as the mall. Now if you live three miles north of the mall and one mile west then, in the y direction it is 3, and the -x direction it is -1. Ellen lives 2 miles south which is the -y direction and is -2, and 5 miles east which is the +x direction and is 5. So in terms of points on the graph we can get you live at (-1,3), and ellen lives at (5,-2). Now we can use a^2+b^2=c^2 by counting the grids if you have it on graph paper or you can do it with the distance formula. (x2-x1)^2+(y2-y1)^2=distance^2. where (x2,y2) are replaced with (5,-2) and (x1,y1) are replaced with (-1,3)

This would make (5-(-1))^2+(3-(-2))^2=distance^2, and when all the math is done 36+25=distance, 61=distance^2, or the squareroot of 61, 61^(1/2)

Is there a different way? I don't understand.

Yes, there is another way to solve this problem using the concept of vectors. A vector is a mathematical object that represents both magnitude and direction. In this problem, we can find the displacement vectors for both you and Ellen, and then calculate the shortest distance between your houses.

Let's consider your location first. You live 3 miles north and 1 mile west of the mall. We can represent this displacement vector as (-1, 3). The negative sign in the x-component indicates that you are moving west (left) from the mall, and the positive y-component indicates that you are moving north.

Now let's consider Ellen's location. She lives 2 miles south and 5 miles east of the mall. We can represent her displacement vector as (5, -2). The positive x-component indicates that she is moving east, and the negative y-component indicates that she is moving south.

To calculate the shortest distance between your houses, we need to find the magnitude of the vector that connects your location to Ellen's location. This can be done using the formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of your location and Ellen's location respectively.

Substituting the values into the formula, we get:

distance = sqrt((5 - (-1))^2 + (-2 - 3)^2)
distance = sqrt((6)^2 + (-5)^2)
distance = sqrt(36 + 25)
distance = sqrt(61)

Therefore, the shortest distance between your houses is sqrt(61) miles.

I hope this explanation helps you understand the alternate method to solve the problem. Let me know if you have any further questions!