Question

find the distance between points.

1. a (-2,2) and b (1,3)

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). For points \( A(-2, 2) \) and \( B(1, 3) \), we have \( x_1 = -2 \), \( y_1 = 2 \), \( x_2 = 1 \), \( y_2 = 3 \).

Step2: Substitute values into formula

Substitute the values into the formula: \( d = \sqrt{(1 - (-2))^2 + (3 - 2)^2} \). Simplify the expressions inside the square root: \( 1 - (-2) = 3 \) and \( 3 - 2 = 1 \). So we get \( d = \sqrt{3^2 + 1^2} \).

Step3: Calculate the squares

Calculate \( 3^2 = 9 \) and \( 1^2 = 1 \). Then the expression becomes \( d = \sqrt{9 + 1} \).

Step4: Sum and take square root

Sum the values inside the square root: \( 9 + 1 = 10 \). So \( d = \sqrt{10} \).

Answer:

\(\sqrt{10}\)