Question

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, the first point (let's say \( (x_1, y_1) \)) is at \( (6, 9) \) and the second point \( (x_2, y_2) \) is at \( (9, 6) \).

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 \( x_1 = 6 \), \( y_1 = 9 \), \( x_2 = 9 \), and \( y_2 = 6 \) into the formula:

\( d = \sqrt{(9 - 6)^2 + (6 - 9)^2} \)

Step3: Simplify the expression inside the square root

Calculate the differences: \( 9 - 6 = 3 \) and \( 6 - 9 = -3 \).

Then square these differences: \( (3)^2 = 9 \) and \( (-3)^2 = 9 \).

Add the squared differences: \( 9 + 9 = 18 \).

So, \( d = \sqrt{18} \).

Step4: Simplify the radical

Simplify \( \sqrt{18} \). We can factor 18 as \( 9 \times 2 \), and \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).

Answer:

\( 3\sqrt{2} \)