Problem 103 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Let \(\quad\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.

Step-by-Step Solution

Answer

The distance is \(3\sqrt{5}\).

Step 1: Identify given points

The task provides us with two points on the cartesian plane: Point 1 \((x_{1}, y_{1}) = (7,2)\), and Point 2 \((x_{2}, y_{2}) = (1,-1)\).

Step 2: Insert the values into the distance formula

We need to substitute these values into the distance formula, which is \(\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\). Doing this gives us \(\sqrt{(1-7)^{2}+(-1-2)^{2}}\).

Step 3: Simplify under the square root

Simplify the expression under the square root. We get \(\sqrt{(-6)^{2}+(-3)^{2}}\), which further simplifies to \(\sqrt{36+9}\).

Step 4: Final simplification

We are left with \(\sqrt{45}\). However, this can still be simplified further. The number 45 is 9*5, and since the square root of 9 is 3, we can simplify \(\sqrt{45}\) to \(3\sqrt{5}\).