Question

the position of two friends sitting in a park on a coordinate plane is shown. the first friend is sitting at point a and the second friend is sitting at point b. if the third friend sits at point c between them, such that the distance between point b and point c is the same as the distance between point a and point c, what are the coordinates of point c? (-6, -4) (9, 6) (1, 2/3) (3, 2) (2, 3/1) (-5, 15/2)

Explanation:

Step1: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Since the distance between point $A(-6,-4)$ and point $C$ is the same as the distance between point $B(9,6)$ and point $C$, point $C$ is the mid - point of the line segment $AB$.

Step2: Calculate the x - coordinate of point C

Let $(x_1,y_1)=(-6,-4)$ and $(x_2,y_2)=(9,6)$. The x - coordinate of the mid - point $x=\frac{x_1 + x_2}{2}=\frac{-6 + 9}{2}=\frac{3}{2}$.

Step3: Calculate the y - coordinate of point C

The y - coordinate of the mid - point $y=\frac{y_1 + y_2}{2}=\frac{-4+6}{2}=1$.

Answer:

$(\frac{3}{2},1)$