Question

given (8, 3) and (x, - 9), find all x such that the distance between these two points is 13. separate multiple answers with a comma

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(8,3)$ and $(x_2,y_2)=(x,- 9)$ and $d = 13$. So, $13=\sqrt{(x - 8)^2+(-9 - 3)^2}$.

Step2: Simplify the equation

First, calculate $(-9 - 3)^2=(-12)^2 = 144$. The equation becomes $13=\sqrt{(x - 8)^2+144}$. Square both sides: $13^2=(x - 8)^2+144$. So, $169=(x - 8)^2+144$.

Step3: Isolate the squared - term

Subtract 144 from both sides: $(x - 8)^2=169 - 144=25$.

Step4: Solve for x

Take the square root of both sides: $x - 8=\pm5$.
Case 1: When $x - 8 = 5$, then $x=5 + 8=13$.
Case 2: When $x - 8=-5$, then $x=-5 + 8 = 3$.

Answer:

$3,13$