Question

find the distance between the two points and the midpoint of the line segment joining them. (1,3) and (7, -5) the distance between the two points is . (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \( x_1 = 1 \), \( y_1 = 3 \), \( x_2 = 7 \), \( y_2 = -5 \).

Step2: Substitute values into formula

First, calculate \( x_2 - x_1 = 7 - 1 = 6 \) and \( y_2 - y_1 = -5 - 3 = -8 \). Then, substitute into the formula: \( d = \sqrt{(6)^2 + (-8)^2} \).

Step3: Simplify the expression

Calculate \( 6^2 = 36 \) and \( (-8)^2 = 64 \). Then, \( 36 + 64 = 100 \). So, \( d = \sqrt{100} = 10 \).

Answer:

10