Question

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

Explanation:

Step1: Identify coordinates of points

First, we identify the coordinates of the two points. From the graph, the first point (let's say \( P \)) is at \( (-1, 7) \) and the second point (let's say \( Q \)) is at \( (8, -5) \).

Step2: Apply 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 \( x_1 = -1 \), \( y_1 = 7 \), \( x_2 = 8 \), \( y_2 = -5 \) into the formula:

\( d = \sqrt{(8 - (-1))^2 + (-5 - 7)^2} \)

Step3: Simplify the expression inside the square root

First, calculate the differences:

\( 8 - (-1) = 8 + 1 = 9 \)

\( -5 - 7 = -12 \)

Then square these differences:

\( 9^2 = 81 \)

\( (-12)^2 = 144 \)

Add these results:

\( 81 + 144 = 225 \) Wait, no, wait: Wait, \( 9^2 = 81 \), \( (-12)^2 = 144 \), so \( 81 + 144 = 225 \)? Wait, no, 81 + 144 is 225? Wait, 81 + 144: 80 + 140 = 220, 1 + 4 = 5, so 225. Wait, but then \( \sqrt{225} = 15 \)? Wait, no, that can't be right. Wait, maybe I misread the coordinates. Wait, let's check again. Wait, the first point: looking at the graph, the x-coordinate: the grid lines. Let's see, the first point is at x = -1? Wait, maybe the first point is at (-1, 7) and the second at (8, -5). Wait, let's recalculate:

\( x_2 - x_1 = 8 - (-1) = 9 \)

\( y_2 - y_1 = -5 - 7 = -12 \)

Then \( (9)^2 + (-12)^2 = 81 + 144 = 225 \), so \( \sqrt{225} = 15 \). Wait, but that seems straightforward. Wait, maybe I made a mistake in coordinates. Wait, let's check the graph again. The first point: on the left side, x = -1 (since it's one unit left of the origin), y = 7 (seven units up). The second point: x = 8 (eight units right), y = -5 (five units down). So the horizontal distance is 8 - (-1) = 9, vertical distance is |-5 - 7| = 12. Then by Pythagoras, the distance is \( \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \). Wait, but 225 is a perfect square, so the simplest radical form is 15.

Wait, but maybe I misread the coordinates. Let me confirm. If the first point is at (-1, 7) and the second at (8, -5), then the differences are 9 and -12, squares are 81 and 144, sum is 225, square root is 15. So that's correct.

Answer:

\( 15 \)