Question

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

Explanation:

Step1: Identify the coordinates

From the graph, the two points are \((-6, 8)\) and \((-1, -4)\). Let \((x_1, y_1)=(-6, 8)\) and \((x_2, y_2)=(-1, -4)\).

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 the values:
\(x_2 - x_1=-1 - (-6)=-1 + 6 = 5\)
\(y_2 - y_1=-4 - 8=-12\)

Step3: Calculate the distance

\(d=\sqrt{(5)^2+(-12)^2}=\sqrt{25 + 144}=\sqrt{169}=13\)? Wait, no, wait: Wait, \((-12)^2 = 144\), \(5^2=25\), \(25 + 144 = 169\), \(\sqrt{169}=13\)? Wait, but let's check the coordinates again. Wait, the lower point: looking at the graph, the lower yellow dot is at \(x=-1\), \(y = - 4\)? Wait, no, maybe I misread the coordinates. Wait, the upper point: x=-6, y=8. The lower point: x=-1, y=-4? Wait, let's re - check the grid. Each grid is 1 unit. So from x=-6 to x=-1: that's 5 units (since -1 - (-6)=5). From y = 8 to y=-4: that's 8 - (-4)=12 units (vertical distance). Then the distance should be \(\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\). Wait, but \(\sqrt{169}=13\), which is an integer, not a radical. But maybe I misread the coordinates. Wait, maybe the lower point is at x=-1, y=-4? Wait, let's check the y - axis. The lower yellow dot is at y=-4, x=-1. The upper one is at x=-6, y=8. So the horizontal difference is \(-1-(-6)=5\), vertical difference is \(-4 - 8=-12\), absolute value 12. Then distance is \(\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\).

Wait, but the problem says "simplest radical form", but 13 is an integer. Maybe I made a mistake in coordinates. Wait, let's re - examine the graph. The upper point: x=-6, y=8 (since it's 6 units left of origin on x - axis, 8 units up on y - axis). The lower point: x=-1, y=-4 (1 unit left of origin on x - axis, 4 units down on y - axis). So the differences: \(\Delta x=-1-(-6)=5\), \(\Delta y=-4 - 8=-12\). Then distance \(d=\sqrt{(5)^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\). So the distance is 13.

Answer:

\(13\)