Point
\[\blue{A}\] is at
\[\blue{(-2, -7)}\] and point
\[\purple{M}\] is at
\[\purple{(2.5, -1.5)}\].
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 can use the midpoint formula. The formula states that if the coordinates of the two endpoints of a segment are $(x_1, y_1)$ and $(x_2, y_2)$, then the coordinates of the midpoint are $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$.
In this problem, the two endpoints of the segment are $\blue{A}$ at $(-2, -7)$ and $\green{B}$ at $(x, y)$. The midpoint $\purple{M}$ is given to be $(2.5, -1.5)$. Using the midpoint formula, we have \[(2.5,-1.5) = \left(\dfrac{x + (-2)}{2}, \dfrac{y + (-7)}{2}\right).\] Simplifying this equation gives us \[(2.5, -1.5) = \left(\dfrac{x - 2}{2}, \dfrac{y - 7}{2}\right).\] We can solve this equation to find that $x-2=5$ and $y-7=-3$. Therefore, $x=7$ and $y=4$.
Hence the coordinates of point $\green{B}$ are $\boxed{(7, 4)}$.
To find the coordinates of point B, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of two points (x1, y1) and (x2, y2) are given by the formula:
\[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \]
Let's use this formula to find the coordinates of point B.
Given:
Point A: (-2, -7)
Point M: (2.5, -1.5)
Using the midpoint formula:
\[ x_B = 2 \times x_M - x_A \]
\[ y_B = 2 \times y_M - y_A \]
Substituting the given values:
\[ x_B = 2 \times 2.5 - (-2) \]
\[ y_B = 2 \times (-1.5) - (-7) \]
Simplifying:
\[ x_B = 5 + 2 \]
\[ x_B = 7 \]
\[ y_B = -3 - 7 \]
\[ y_B = -10 \]
Therefore, the coordinates of point B are (7, -10).
To find the coordinates of point \(\green{B}\), we can use the midpoint formula. The midpoint formula states that if the coordinates of two points, \(A\) and \(B\), are given, then the coordinates of the midpoint, \(M\), between them can be found by taking the average of their respective coordinates.
Let's use this formula to find the coordinates of point \(\green{B}\):
Step 1: Given that point \(\purple{M}\) is the midpoint between point \(\blue{A}\) and point \(\green{B}\), we can set up the equation:
\[\frac{{\blue{x_A} + \green{x_B}}}{2} = \purple{x_M}\]
\[\frac{{\blue{y_A} + \green{y_B}}}{2} = \purple{y_M}\]
Step 2: Substitute the given coordinates into the equation:
\[\frac{{\blue{-2} + \green{x_B}}}{2} = \purple{2.5}\]
\[\frac{{\blue{-7} + \green{y_B}}}{2} = \purple{-1.5}\]
Step 3: Solve for the unknowns.
For the x-coordinate:
\[\frac{{-2 + \green{x_B}}}{2} = 2.5\]
Multiply both sides by 2:
\[-2 + \green{x_B} = 5\]
Add 2 to both sides:
\[\green{x_B} = 7\]
For the y-coordinate:
\[\frac{{-7 + \green{y_B}}}{2} = -1.5\]
Multiply both sides by 2:
\[-7 + \green{y_B} = -3\]
Add 7 to both sides:
\[\green{y_B} = 4\]
Therefore, the coordinates of point \(\green{B}\) are \((7, 4)\).