To answer your questions, we will use the distance formula and the midpoint formula.
Question 1
Distance formula: The distance \( d \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points \( A(-3, 5) \) and \( B(7, 2) \):
- \( x_1 = -3 \)
- \( y_1 = 5 \)
- \( x_2 = 7 \)
- \( y_2 = 2 \)
Now, plug in the values into the formula: \[ d = \sqrt{(7 - (-3))^2 + (2 - 5)^2} \] \[ = \sqrt{(7 + 3)^2 + (2 - 5)^2} \] \[ = \sqrt{(10)^2 + (-3)^2} \] \[ = \sqrt{100 + 9} \] \[ = \sqrt{109} \] Calculating \( \sqrt{109} \): \[ \sqrt{109} \approx 10.4403 \] Rounding to the nearest tenth, the distance is approximately: \[ \text{Distance} \approx 10.4 \text{ units} \]
Question 2
Midpoint formula: The midpoint \( M \) of a line segment between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points \( A(-3, 5) \) and \( B(7, 2) \): \[ M = \left( \frac{-3 + 7}{2}, \frac{5 + 2}{2} \right) \] \[ = \left( \frac{4}{2}, \frac{7}{2} \right) \] \[ = (2, 3.5) \]
Final Answers
- The distance from point A to point B is approximately 10.4 units.
- The midpoint of line segment AB is (2, 3.5).
4 answers