Question

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

Explanation:

1. First, assume the two - point formula:

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Since the points are not given explicitly in the text, let's assume the two points are \((x_1,y_1)\) and \((x_2,y_2)\).
• However, if we assume one point is \((0,5)\) (from the \(y\) - axis at \(y = 5\)) and the other point is \((- 5,-7)\) (estimated from the grid), then:
• Here \(x_1=0,y_1 = 5,x_2=-5,y_2=-7\).

1. Then, calculate the differences:

• Calculate \(x_2 - x_1\) and \(y_2 - y_1\):
• \(x_2 - x_1=-5 - 0=-5\).
• \(y_2 - y_1=-7 - 5=-12\).

1. Next, apply the distance formula:

• Substitute into the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• \(d=\sqrt{(-5)^2+(-12)^2}=\sqrt{25 + 144}\).
• \(d=\sqrt{169}\).
• \(d = 13\) (but we need it in radical form, and \(\sqrt{169}=13\) can also be written as \(\sqrt{13\times13}\)). In simplest radical form, \(d = 13\) (since \(\sqrt{169}\) is already in a simple form considering we are looking for the square - root result). If we assume other points, say \((x_1,y_1)\) and \((x_2,y_2)\):
• \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's start over with general steps.
• Let the two points be \((x_1,y_1)\) and \((x_2,y_2)\).
• First, find the difference in \(x\) - coordinates: \(\Delta x=x_2 - x_1\) and the difference in \(y\) - coordinates: \(\Delta y=y_2 - y_1\).
• Then \(d=\sqrt{(\Delta x)^2+(\Delta y)^2}\).
• Suppose the two points are \((0,5)\) and \((-5,-7)\):
• \(\Delta x=-5-0=-5\), \(\Delta y=-7 - 5=-12\).
• \(d=\sqrt{(-5)^2+(-12)^2}=\sqrt{25+144}=\sqrt{169}=13=\sqrt{13^2}\).

Answer:

\(\sqrt{169}\) (or \(13\) written as a square - root \(\sqrt{13^2}\), and the simplest radical form of \(\sqrt{169}\) is \(13\) but if we must keep it in radical form \(\sqrt{169}\) is correct as \(169 = 13\times13\))