Find the distance between the pair of points. Give an exact answer and an approximation. \( (3,5) \) and \( (4,10) \)

To find the distance between the points \( (3,5) \) and \( (4,10) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. 1. Identify the coordinates: - \( (x_1, y_1) = (3, 5) \) - \( (x_2, y_2) = (4, 10) \) 2. Substitute the values into the distance formula: \[ d = \sqrt{(4 - 3)^2 + (10 - 5)^2} \] 3. Calculate the differences: - \( x_2 - x_1 = 4 - 3 = 1 \) - \( y_2 - y_1 = 10 - 5 = 5 \) 4. Substitute these differences back into the formula: \[ d = \sqrt{(1)^2 + (5)^2} \] 5. Calculate the squares: \[ d = \sqrt{1 + 25} \] 6. Add the values: \[ d = \sqrt{26} \] Now, we have the exact distance as \( \sqrt{26} \). To find the approximate distance, we can calculate the square root of 26. Let's perform the calculation for the approximation. \[ \sqrt{26} \approx 5.099 \] Thus, the exact distance between the points \( (3,5) \) and \( (4,10) \) is \( \sqrt{26} \), and the approximate distance is \( 5.099 \).