Question

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

Explanation:

Step1: Identify the coordinates of the two points

From the graph, the two points are \((-1,7)\) and \((7,-5)\). Let \((x_1,y_1)=(-1,7)\) and \((x_2,y_2)=(7,-5)\).

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

Substitute the values:

$$
LATEXBLOCK0
$$

Step3: Calculate the squares

$$
LATEXBLOCK1
$$

Step4: Simplify the radical

Factor \(208\): \(208=16\times13\). Then \(\sqrt{208}=\sqrt{16\times13}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 16\), \(b = 13\)), we get \(\sqrt{16}\cdot\sqrt{13}=4\sqrt{13}\).

Answer:

\(4\sqrt{13}\)