Point
\[\blue{A}\] is at
\[\blue{(-6, 8)}\] and point
\[\purple{M}\] is at
\[\purple{(0, 0.5)}\].
Point
\[\purple{M}\] is the midpoint of point
\[\blue{A}\] and point
\[\green{B}\].
What are the coordinates of point
\[\green{B}\]?
Since point \[\purple{M}\] is the midpoint of point \[\blue{A}\] and point \[\green{B}\], the coordinates of point \[\green{B}\] can be found by subtracting the difference between the x-coordinates of point \[\purple{M}\] and point \[\blue{A}\] from the x-coordinate of point \[\purple{M}\], and by subtracting the difference between the y-coordinates of point \[\purple{M}\] and point \[\blue{A}\] from the y-coordinate of point \[\purple{M}\].
The x-coordinate of point \[\purple{M}\] is \[0\], and the x-coordinate of point \[\blue{A}\] is \[-6\]. So, the difference between the x-coordinates is \[0 - (-6) = 6\].
The y-coordinate of point \[\purple{M}\] is \[0.5\], and the y-coordinate of point \[\blue{A}\] is \[8\]. So, the difference between the y-coordinates is \[0.5 - 8 = -7.5\].
Applying these differences to the x and y coordinates of \[\purple{M}\], we get
\[\green{B} = (0 - 6, 0.5 - 7.5) = \boxed{(-6, -7)}. \]
Point
\[\blue{A}\] is at
\[\blue{(-6, 8)}\] and point
\[\purple{M}\] is at
\[\purple{(0, 0.5)}\].
Point
\[\purple{M}\] is the midpoint of point
\[\blue{A}\] and point
\[\green{B}\].
What are the coordinates of point
\[\green{B}\]?
To find the coordinates of point \(\green{B}\), we first need to understand that the midpoint of two points is the average of their coordinates. In this case, point \(\purple{M}\) is the midpoint of point \(\blue{A}\) and point \(\green{B}\).
So, we can use the midpoint formula to find the coordinates of point \(\green{B}\). The midpoint formula is given by:
\[\text{{Midpoint}} = \left(\dfrac{{x_1 + x_2}}{2}, \dfrac{{y_1 + y_2}}{2}\right)\]
We know that point \(\purple{M}\) has coordinates \(\purple{(0, 0.5)}\) and point \(\blue{A}\) has coordinates \(\blue{(-6, 8)}\). Let's substitute these values into the midpoint formula:
\[\left(\dfrac{{-6 + x_2}}{2}, \dfrac{{8 + y_2}}{2}\right) = \purple{(0, 0.5)}\]
From this equation, we can solve for \(x_2\) and \(y_2\), which represent the coordinates of point \(\green{B}\).
Solving for \(x_2\):
\[\dfrac{{-6 + x_2}}{2} = 0 \implies -6 + x_2 = 0 \implies x_2 = 6\]
Solving for \(y_2\):
\[\dfrac{{8 + y_2}}{2} = 0.5 \implies 8 + y_2 = 1 \implies y_2 = 1-8 = -7\]
Therefore, the coordinates of point \(\green{B}\) are \(\green{(6, -7)}\).
To find the coordinates of point \[\green{B}\], we can use the midpoint formula.
The midpoint formula states that the midpoint of two points, \[\blue{(x_1, y_1)}\] and \[\green{(x_2, y_2)}\], is given by the coordinates \[\purple{\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)}\].
In this case, we know the coordinates of point \[\blue{A (-6, 8)}\] and the midpoint \[\purple{M(0, 0.5)}\].
Let's substitute the values into the midpoint formula:
\[\purple{x_M =\frac{{x_A + x_B}}{2}}\] and \[\purple{y_M = \frac{{y_A + y_B}}{2}}\].
Substituting the known values, we get:
\[\purple{0 = \frac{{-6 + x_B}}{2}}\] and \[\purple{0.5 = \frac{{8 + y_B}}{2}}\].
To isolate \[\purple{x_B}\], we can multiply both sides of the first equation by 2:
\[2 \cdot 0 = -6 + x_B \Rightarrow 0 = -6 + x_B\].
Adding 6 to both sides gives us:
\[x_B = 6\].
Similarly, to isolate \[\purple{y_B}\], we can multiply both sides of the second equation by 2:
\[2 \cdot 0.5 = 8 + y_B \Rightarrow 1 = 8 + y_B\].
Subtracting 8 from both sides gives us:
\[y_B = -7\].
Therefore, the coordinates of point \[\green{B}\] are \[\green{(6, -7)}\].
Since point \[\purple{M}\] is the midpoint of point \[\blue{A}\] and point \[\green{B}\], the coordinates of point \[\green{B}\] can be found by adding the difference between the x-coordinates of point \[\purple{M}\] and point \[\blue{A}\] to the x-coordinate of point \[\purple{M}\], and by adding the difference between the y-coordinates of point \[\purple{M}\] and point \[\blue{A}\] to the y-coordinate of point \[\purple{M}\].
The x-coordinate of point \[\purple{M}\] is \[0\], and the x-coordinate of point \[\blue{A}\] is \[-6\]. So, the difference between the x-coordinates is \[0 - (-6) = 6\].
The y-coordinate of point \[\purple{M}\] is \[0.5\], and the y-coordinate of point \[\blue{A}\] is \[8\]. So, the difference between the y-coordinates is \[0.5 - 8 = -7.5\].
Applying these differences to the x and y coordinates of \[\purple{M}\], we get
\[\green{B} = (0 + 6, 0.5 + (-7.5)) = \boxed{(6, -7)}.\]