Calculate the distance between the points \( (3, 4) \) and \( (7, 1) \).

To calculate the distance between two points in a Cartesian plane, you can use the **Distance Formula**, which is derived from the Pythagorean theorem. Given two points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \), the distance \( d \) between them is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Let's apply this formula to your specific points \( (3, 4) \) and \( (7, 1) \). 1. **Identify the coordinates:** - \( P_1 = (3, 4) \) where \( x_1 = 3 \) and \( y_1 = 4 \) - \( P_2 = (7, 1) \) where \( x_2 = 7 \) and \( y_2 = 1 \) 2. **Subtract the corresponding coordinates:** - \( x_2 - x_1 = 7 - 3 = 4 \) - \( y_2 - y_1 = 1 - 4 = -3 \) 3. **Square these differences:** - \( (x_2 - x_1)^2 = 4^2 = 16 \) - \( (y_2 - y_1)^2 = (-3)^2 = 9 \) 4. **Add the squared differences:** - \( 16 + 9 = 25 \) 5. **Take the square root of the sum to find the distance:** - \( d = \sqrt{25} = 5 \) **Therefore, the distance between the points \( (3, 4) \) and \( (7, 1) \) is 5 units.**