Point
\[\blue{A}\] is at
\[\blue{(-5, -4)}\] and point
\[\purple{M}\] is at
\[\purple{(0, -3.5)}\].
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 \(\green{B}\), we can use the midpoint formula to find the coordinates of point \(\green{B}\). The midpoint formula states that if the coordinates of two points are \((x_1, y_1)\) and \((x_2, y_2)\), then the coordinates of their midpoint are \(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\).
In this case, the coordinates of point \(\purple{M}\) are \((0, -3.5)\) and the coordinates of point \(\blue{A}\) are \((-5, -4)\). Let the coordinates of point \(\green{B}\) be \((x, y)\). Using the midpoint formula, we have:
\[\left(\frac{{-5 + x}}{2}, \frac{{-4 + y}}{2}\right) = (0, -3.5)\]
Simplifying the equation, we get:
\[\frac{{-5 + x}}{2} = 0 \ \ \ \Rightarrow \ \ \ x = 5\]
\[\frac{{-4 + y}}{2} = -3.5 \ \ \ \Rightarrow \ \ \ -4 + y = -7 \ \ \ \Rightarrow \ \ \ y = -3\]
Therefore, the coordinates of point \(\green{B}\) are \((5, -3)\).
Point
\[\blue{A}\] is at
\[\blue{(-5, -4)}\] and point
\[\purple{M}\] is at
\[\purple{(0, -3.5)}\].
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 midpoint formula states that the coordinates of the midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the following formulas:
\[x_m = \frac{{x_1 + x_2}}{2}\]
\[y_m = \frac{{y_1 + y_2}}{2}\]
In this case, the coordinates of point \(\blue{A}\) are \((-5, -4)\) and the coordinates of point \(\purple{M}\) are \((0, -3.5)\). We know that point \(\purple{M}\) is the midpoint of points \(\blue{A}\) and \(\green{B}\). Using the midpoint formula, we can substitute the given coordinates into the formulas to find the coordinates of point \(\green{B}\).
Let's calculate the \(x\)-coordinate of point \(\green{B}\):
\[x_m = \frac{{x_1 + x_2}}{2}\]
\[0 = \frac{{-5 + x_2}}{2}\]
To solve for \(x_2\), we can multiply both sides of the equation by 2:
\[0 = -5 + x_2\]
Adding 5 to both sides of the equation gives:
\[5 = x_2\]
So, the \(x\)-coordinate of point \(\green{B}\) is \(5\).
Now, let's calculate the \(y\)-coordinate of point \(\green{B}\):
\[y_m = \frac{{y_1 + y_2}}{2}\]
\[-3.5 = \frac{{-4 + y_2}}{2}\]
To solve for \(y_2\), we can multiply both sides of the equation by 2:
\[-7 = -4 + y_2\]
Adding 4 to both sides of the equation gives:
\[-3 = y_2\]
So, the \(y\)-coordinate of point \(\green{B}\) is \(-3\).
Therefore, the coordinates of point \(\green{B}\) are \((5, -3)\).
Since point \(\purple{M}\) is the midpoint of point \(\blue{A}\) and point \(\green{B}\), we can use the midpoint formula to find the coordinates of point \(\green{B}\). The midpoint formula states that if the coordinates of two points are \((x_1, y_1)\) and \((x_2, y_2)\), then the coordinates of their midpoint are \(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\).
In this case, the coordinates of point \(\purple{M}\) are \((0, -3.5)\) and the coordinates of point \(\blue{A}\) are \((-5, -4)\). Let the coordinates of point \(\green{B}\) be \((x, y)\). Using the midpoint formula, we have:
\[ \left(\frac{{-5 + x}}{2}, \frac{{-4 + y}}{2}\right) = (0, -3.5)\]
Simplifying the equation, we get:
\[\frac{{-5 + x}}{2} = 0\]
\[-5 + x = 0\]
\[x = 5\]
\[\frac{{-4 + y}}{2} = -3.5\]
\[-4 + y = -7\]
\[y = -3\]
Therefore, the coordinates of point \(\green{B}\) are \((5, -3)\).
To find the coordinates of point \(\green{B}\), we can use the midpoint formula.
The midpoint formula states that if a point \((x_1, y_1)\) is the midpoint of two other points \((x_2, y_2)\) and \((x_3, y_3)\), then the coordinates of \((x_2, y_2)\) and \((x_3, y_3)\) can be found using the following formulas:
\[x_2 = 2x_1 - x_3\]
\[y_2 = 2y_1 - y_3\]
In this case, we know that point \(\purple{M}\) is the midpoint of points \(\blue{A}\) and \(\green{B}\). We are given the coordinates of point \(\blue{A}\) as \((-5, -4)\) and the coordinates of point \(\purple{M}\) as \((0, -3.5)\).
Let's substitute these values into the formulas:
\[x_2 = 2 \cdot 0 - x_3\]
\[y_2 = 2 \cdot (-3.5) - y_3\]
Simplifying these equations, we get:
\[x_2 = -x_3\]
\[y_2 = -7 - y_3\]
Since point \(\green{B}\) has coordinates \((x_2, y_2)\), we can rewrite these equations as:
\[x_2 = -x_3\]
\[y_2 = -7 - y_3\]
To solve for point \(\green{B}\), we need to find values for \(x_2\) and \(y_2\). From the given information, we know that point \(\purple{M}\) is at \((0, -3.5)\), which means that \(x_3 = 0\) and \(y_3 = -3.5\).
Substituting these values into the equations, we get:
\[x_2 = -0 = 0\]
\[y_2 = -7 - (-3.5) = -7 + 3.5 = -3.5\]
Therefore, the coordinates of point \(\green{B}\) are \((0, -3.5)\).