To find the distance between Carmen's house and the store, you can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For Carmen's house at (8, 2) and the store at (7, 5):
- \(x_1 = 8\), \(y_1 = 2\)
- \(x_2 = 7\), \(y_2 = 5\)
Plugging in the coordinates:
\[ d = \sqrt{(7 - 8)^2 + (5 - 2)^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
So, the distance between Carmen's house and the store is \(\sqrt{10}\), which is approximately 3.16 units.
Next, to find the midpoint between Carmen's house and school, you can use the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For Carmen's house at (8, 2) and school at (4, 4):
- \(x_1 = 8\), \(y_1 = 2\)
- \(x_2 = 4\), \(y_2 = 4\)
Calculating the midpoint:
\[ \text{Midpoint} = \left( \frac{8 + 4}{2}, \frac{2 + 4}{2} \right) = \left( \frac{12}{2}, \frac{6}{2} \right) = (6, 3) \]
Thus, Carmen stopped at the coordinates (6, 3), which is the location of the gym.