In the standard (x, y) coordinate plane, the midpoint of `bar (AB)` is (8, -5) and B is located at (12 , -1). If (x, y) are the coordinates of A, what is the value of x + y ?

Options

A. `-8`
B. `-7`
C. `-5`
D. 7

Answer

To solve the problem step by step, we need to find the coordinates of point A given the midpoint of segment AB and the coordinates of point B.

### Step 1: Understand the midpoint formula
The midpoint \( M \) of a line segment connecting points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
In this case, we know the midpoint \( M \) is (8, -5) and the coordinates of point B are (12, -1).

### Step 2: Set up the equations
Let the coordinates of point A be \( (x, y) \). According to the midpoint formula, we can set up the following equations:
1. For the x-coordinates:
\[ \frac{x + 12}{2} = 8 \]
2. For the y-coordinates:
\[ \frac{y - 1}{2} = -5 \]

### Step 3: Solve for x
From the first equation:
\[ \frac{x + 12}{2} = 8 \]
Multiply both sides by 2 to eliminate the fraction:
\[ x + 12 = 16 \]
Now, subtract 12 from both sides:
\[ x = 16 - 12 = 4 \]

### Step 4: Solve for y
From the second equation:
\[ \frac{y - 1}{2} = -5 \]
Multiply both sides by 2:
\[ y - 1 = -10 \]
Now, add 1 to both sides:
\[ y = -10 + 1 = -9 \]

### Step 5: Find x + y
Now that we have the coordinates of point A as \( (4, -9) \), we can find \( x + y \):
\[ x + y = 4 + (-9) = 4 - 9 = -5 \]

### Final Answer
The value of \( x + y \) is \( -5 \).

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