Question

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

Explanation:

Step1: Identify the coordinates

First, we find the coordinates of the two points. From the graph, the first point is at \((1, 2)\) and the second point is at \((7, -6)\).

Step2: Apply the distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Substituting \(x_1 = 1\), \(y_1 = 2\), \(x_2 = 7\), and \(y_2 = -6\) into the formula:
\[

$$\begin{align*} d&=\sqrt{(7 - 1)^2 + (-6 - 2)^2}\\ &=\sqrt{(6)^2 + (-8)^2}\\ &=\sqrt{36 + 64}\\ &=\sqrt{100} \end{align*}$$

\]
Wait, no, wait. Wait, \(7 - 1 = 6\), \(-6 - 2 = -8\). Then \((6)^2 = 36\), \((-8)^2 = 64\). Then \(36 + 64 = 100\)? Wait, no, that can't be. Wait, maybe I misread the coordinates. Wait, let's check again. Wait, the first point: x=1, y=2. The second point: x=7, y=-6? Wait, no, looking at the graph, the second point is at (7, -6)? Wait, no, the grid: let's count the x and y. Wait, the first point is at (1, 2) (x=1, y=2). The second point: x=7, y=-6? Wait, no, maybe I made a mistake. Wait, let's recalculate. Wait, \(x_2 - x_1 = 7 - 1 = 6\), \(y_2 - y_1 = -6 - 2 = -8\). Then squared terms: \(6^2 = 36\), \((-8)^2 = 64\). Sum: \(36 + 64 = 100\). Then square root of 100 is 10? Wait, but that seems too simple. Wait, maybe the coordinates are different. Wait, let's check the graph again. Wait, the first point: x=1, y=2. The second point: x=7, y=-6? Wait, no, maybe the second point is at (7, -6)? Wait, the y-coordinate: from y=2 to y=-6, that's a difference of -8. x-coordinate: 1 to 7, difference of 6. Then distance is \(\sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\). Wait, but that's a whole number. Maybe that's correct.

Wait, maybe I misread the second point. Let me check again. The second point is at (7, -6)? Let's see the grid. The x-axis: from 1 to 7, that's 6 units. The y-axis: from 2 to -6, that's 8 units down. So the horizontal distance is 6, vertical distance is 8. Then it's a right triangle with legs 6 and 8, so hypotenuse is 10 (since 6-8-10 is a Pythagorean triple). So that's correct.

Answer:

The distance between the two points is \(\boxed{10}\). Wait, but the problem says "simplest radical form". Wait, but \(\sqrt{100} = 10\), which is an integer, so that's the simplest form. So the answer is 10.