Question

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

Explanation:

Step1: Identify the coordinates

The first point (yellow dot on the upper left) is at \((-6, 8)\) and the second point (yellow dot on the lower middle) is at \((-1, -4)\).

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 \(x_1 = -6\), \(y_1 = 8\), \(x_2 = -1\), \(y_2 = -4\) into the formula:

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Wait, no, wait. Wait, \(-4 - 8 = -12\), squared is 144. \(-1 - (-6) = 5\), squared is 25. 25 + 144 = 169. Square root of 169 is 13? Wait, but let me check the coordinates again. Wait, the lower point: looking at the graph, the x-coordinate is -1? Wait, no, the lower yellow dot: x is -1? Wait, the grid: from x=-1, y=-4? Wait, maybe I misread the coordinates. Wait, let's re-express the coordinates.

Wait, the upper point: x = -6, y = 8 (since it's at x=-6, y=8). The lower point: x = -1? Wait, no, looking at the x-axis: the lower yellow dot is at x=-1? Wait, the x-axis labels: -10, -9, ..., -1, 0, 1, ..., 10. So the lower yellow dot is at x=-1, y=-4? Wait, no, maybe I made a mistake. Wait, let's count the grid. The upper point: x=-6, y=8. The lower point: let's see, x: from -6 to -1 is 5 units (since -1 - (-6) = 5). y: from 8 to -4 is 12 units (8 - (-4) = 12). So the horizontal distance is 5, vertical distance is 12. Then distance is \(\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). Wait, but that's a whole number. But maybe I misread the coordinates. Wait, let's check again.

Wait, the lower point: x-coordinate: looking at the graph, the lower yellow dot is at x=-1? Wait, the x-axis: the grid lines are at each integer. So the lower point is at x=-1, y=-4? Wait, no, maybe the lower point is at x=-1, y=-4? Wait, let's confirm:

Upper point: (x, y) = (-6, 8) (since it's 6 units left of the origin on x, 8 up on y).

Lower point: (x, y) = (-1, -4) (1 unit left of origin on x, 4 down on y).

So the difference in x: \(-1 - (-6) = 5\)

Difference in y: \(-4 - 8 = -12\)

Then distance: \(\sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). Wait, but 5-12-13 is a Pythagorean triple. So that's correct.

Wait, but maybe I misread the lower point's x-coordinate. Wait, maybe the lower point is at x=-1? Let me check the graph again. The lower yellow dot: x is -1, y is -4. Yes, because it's at the intersection of x=-1 and y=-4. The upper yellow dot is at x=-6, y=8. So the horizontal change is 5 (from -6 to -1: 5 units right), vertical change is 12 (from 8 to -4: 12 units down). So the distance is \(\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). Wait, but that seems straightforward. But let me confirm the coordinates again.

Alternatively, maybe the lower point is at x=-1, y=-4. So the distance formula gives 13. Wait, but the problem says "simplest radical form", but 13 is an integer, so that's the simplest form.

Answer:

\(\boxed{13}\)