To find the coordinates of point C given that point B (6, 0) is the midpoint of point A (2, 3) and point C (?, ?), we need to use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}\][/tex]
Given:
- Point A: [tex]\((2, 3)\)[/tex]
- Point B (midpoint): [tex]\((6, 0)\)[/tex]
- Point C: [tex]\((C_x, C_y)\)[/tex] (unknown)
Since point B is the midpoint, we can set up the following equations:
[tex]\[6 = \frac{2 + C_x}{2}\][/tex]
[tex]\[0 = \frac{3 + C_y}{2}\][/tex]
To find [tex]\(C_x\)[/tex], we solve the first equation:
[tex]\[6 = \frac{2 + C_x}{2}\][/tex]
Multiplying both sides by 2 to clear the fraction:
[tex]\[12 = 2 + C_x\][/tex]
Subtracting 2 from both sides:
[tex]\[C_x = 10\][/tex]
To find [tex]\(C_y\)[/tex], we solve the second equation:
[tex]\[0 = \frac{3 + C_y}{2}\][/tex]
Multiplying both sides by 2 to clear the fraction:
[tex]\[0 = 3 + C_y\][/tex]
Subtracting 3 from both sides:
[tex]\[C_y = -3\][/tex]
So, the coordinates for point C are:
[tex]\[C = (10, -3)\][/tex]
Visualizing the points A, B, and C, you would have point A at [tex]\((2, 3)\)[/tex], point B at [tex]\((6, 0)\)[/tex], and point C at [tex]\((10, -3)\)[/tex]. These points are collinear, with B positioned exactly halfway between A and C.
Therefore, the coordinates for point C are:
[tex]\[
(10, -3)
\][/tex]
