Question

1. find the distance between the two points in simplest radical form.

Explanation:

Step1: Identify the coordinates

First, we identify the coordinates of the two points from the graph. Let's call the first point \( (x_1, y_1) \) and the second point \( (x_2, y_2) \). From the graph, the first point is at \( (1, 5) \) and the second point is at \( (3, 3) \).

Step2: Apply the distance formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Substitute \( x_1 = 1 \), \( y_1 = 5 \), \( x_2 = 3 \), and \( y_2 = 3 \) into the formula:

\( d = \sqrt{(3 - 1)^2 + (3 - 5)^2} \)

Step3: Simplify the expression

First, calculate the differences inside the parentheses:

\( 3 - 1 = 2 \) and \( 3 - 5 = -2 \)

Then, square these differences:

\( (2)^2 = 4 \) and \( (-2)^2 = 4 \)

Add these squared values:

\( 4 + 4 = 8 \)

So, \( d = \sqrt{8} \)

Step4: Simplify the radical

Simplify \( \sqrt{8} \) by factoring out the perfect square:

\( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \)

Answer:

\( 2\sqrt{2} \)