Step 1 :Given points $P_{1}(3,5)$ and $P_{2}(8,-15)$
Step 2 :The distance between two points in a 2D plane can be calculated using the distance formula derived from the Pythagorean theorem. The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Step 3 :Substitute $x_1=3$, $y_1=5$, $x_2=8$, and $y_2=-15$ into the formula
Step 4 :Calculate the result to get the distance
Step 5 :Final Answer: The distance between points $P_{1}(3,5)$ and $P_{2}(8,-15)$ is $\boxed{20.615528128088304}$