Question

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

Explanation:

Step1: Identify the coordinates of the two points

Let the two points be \(A(x_1,y_1)\) and \(B(x_2,y_2)\). From the graph, \(A(1,1)\) and \(B(-7,-3)\).

Step2: Apply the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Substitute \(x_1 = 1,y_1 = 1,x_2=-7,y_2 = -3\) into the formula:

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Step3: Simplify the expression inside the square - root

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Step4: Simplify the radical

Factor \(80\) as \(16\times5\), then \(\sqrt{80}=\sqrt{16\times5}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}(a = 16\geq0,b = 5\geq0)\), we get \(\sqrt{16\times5}=4\sqrt{5}\)

Answer:

\(4\sqrt{5}\)