Question

You and a friend want to meet halfway between your two homes to have coffee. You plot both of your houses on a map and assign coordinates to your locations. Your house is located at point (6 , 4). Your friend's is located at point (-2 , -6). Using the midpoint formula, calculate the halfway point between your two homes. Show all of your work. Showing your work is worth 1 point; the correct answer is worth 1 point. (2 points)

Answer 1

To find the halfway point between two locations on a coordinate plane, we can use the midpoint formula. The formula for the midpoint \( M \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

In this scenario, your coordinates are:

• Your house: \( (6, 4) \) which gives \( x_1 = 6 \) and \( y_1 = 4 \)
• Your friend's house: \( (-2, -6) \) which gives \( x_2 = -2 \) and \( y_2 = -6 \)

Now we can plug these values into the midpoint formula:

1. Calculate the x-coordinate of the midpoint: \[ x_m = \frac{x_1 + x_2}{2} = \frac{6 + (-2)}{2} = \frac{6 - 2}{2} = \frac{4}{2} = 2 \]

2. Calculate the y-coordinate of the midpoint: \[ y_m = \frac{y_1 + y_2}{2} = \frac{4 + (-6)}{2} = \frac{4 - 6}{2} = \frac{-2}{2} = -1 \]

Putting this together, the coordinates of the halfway point (midpoint) between your two homes is: \[ M = (2, -1) \]

So, the halfway point between your homes is \( (2, -1) \).