Question

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

Explanation:

Step1: Identify Coordinates

First, find the coordinates of the two points. Let's assume the first point (let's call it \( A \)) is at \( (1, -1) \) (from the graph: x=1, y=-1) and the second point ( \( B \)) is at \( (-5, -9) \) (x=-5, y=-9).

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 \( x_1 = 1 \), \( y_1 = -1 \), \( x_2 = -5 \), \( y_2 = -9 \):

\( x_2 - x_1 = -5 - 1 = -6 \)

\( y_2 - y_1 = -9 - (-1) = -9 + 1 = -8 \)

Step3: Calculate Squares and Sum

Square the differences: \( (-6)^2 = 36 \), \( (-8)^2 = 64 \)

Sum them: \( 36 + 64 = 100 \)

Step4: Take Square Root

Take the square root: \( d = \sqrt{100} = 10 \)? Wait, no, wait. Wait, maybe I misread the coordinates. Wait, let's recheck. Wait, the first point: looking at the graph, the first yellow dot is at (1, -1)? Wait, no, the x-axis: the first dot is at x=1 (since it's on the first vertical line after 0), y=-1 (below x-axis, first horizontal line). The second dot: x=-5 (since it's 5 units left of 0), y=-9 (9 units down). Wait, but let's recalculate:

\( x_2 - x_1 = -5 - 1 = -6 \), \( y_2 - y_1 = -9 - (-1) = -8 \). Then \( (-6)^2 = 36 \), \( (-8)^2 = 64 \), sum is 100, square root is 10. Wait, but maybe the coordinates are different. Wait, maybe the first point is (1, -1) and the second is (-5, -9). Wait, but let's check the grid. Each square is 1 unit. So from (1, -1) to (-5, -9): horizontal distance is 1 - (-5) = 6? Wait, no, distance is absolute. So horizontal change: \( |1 - (-5)| = 6 \), vertical change: \( |-1 - (-9)| = 8 \). Then distance is \( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \). Wait, that's correct. So the distance is 10. Wait, but the problem says "simplest radical form", but 100 is a perfect square, so sqrt(100)=10.

Wait, maybe I made a mistake in coordinates. Let's re-express the points. Let's denote Point 1: (1, -1) (yellow dot at x=1, y=-1). Point 2: (-5, -9) (yellow dot at x=-5, y=-9). Then the distance formula:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

\( x_1 = 1 \), \( y_1 = -1 \); \( x_2 = -5 \), \( y_2 = -9 \)

\( x_2 - x_1 = -5 - 1 = -6 \)

\( y_2 - y_1 = -9 - (-1) = -8 \)

\( (-6)^2 = 36 \), \( (-8)^2 = 64 \)

Sum: 36 + 64 = 100

Square root: \( \sqrt{100} = 10 \)

So the distance is 10.

Answer:

\( \boxed{10} \)