Question

question
find the distance between the two points in simplest radical form.
(5,2) and (-3,-5)

Explanation:

Step1: Identify coordinates

Let \((x_1,y_1)=(5,2)\) and \((x_2,y_2)=(-3,-5)\).

Step2: Apply distance formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Substitute values: \(d=\sqrt{(-3 - 5)^2+(-5 - 2)^2}\).

Step3: Calculate differences

\(-3-5=-8\), \(-5 - 2=-7\).
So \(d=\sqrt{(-8)^2+(-7)^2}\).

Step4: Compute squares

\((-8)^2 = 64\), \((-7)^2 = 49\).
Then \(d=\sqrt{64 + 49}\).

Step5: Sum and simplify

\(64+49 = 113\).
Since \(113\) is prime, \(d=\sqrt{113}\).

Answer:

\(\sqrt{113}\)