Question

What is the exact distance between $(10, 6)$ and $(1, -4)$?

Answer 1

Answer:

$\boxed{\sqrt{181}}$

Step-by-step explanation:

The formula for the distance between two points is

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

x₂ - x₁ = 10 -   1   =  9

y₂ - y₁ =  6 - (-4) = 10

The formula, in effect, creates a right triangle, so we can use the Pythagorean theorem to calculate the distance.

$\begin{array}{rcl}d & = &\sqrt{9^{2} + 10^{2}}\\& = & \sqrt{81 + 100}\\& = & \sqrt{181}\\\end{array}\\\text{181 is a prime number. It has no factors that allow us to simplify its square root.}\\\text{The exact distance between the points is \boxed{\mathbf{\sqrt{181}}} }$