Point P is the midpoint of segment AB. Co- ordinates of point P are (2,1) and that of point A are (11,5) . The co-ordinates of point B are

Options

A. `(- 7, -3)`
B. (6,5,3)
C. (7,3)
D. (-6, 5, -3)

Answer

To find the coordinates of point B, given that point P is the midpoint of segment AB, we can use the midpoint formula. The midpoint P of a segment AB is calculated as follows:

\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Where:
- \( (x_1, y_1) \) are the coordinates of point A,
- \( (x_2, y_2) \) are the coordinates of point B,
- \( (x, y) \) are the coordinates of point P.

### Step 1: Identify the known values
We know:
- Coordinates of point P: \( (2, 1) \)
- Coordinates of point A: \( (11, 5) \)

Let:
- \( (x_1, y_1) = (11, 5) \) (Coordinates of point A)
- \( (x, y) = (2, 1) \) (Coordinates of point P)
- \( (x_2, y_2) \) = (unknown coordinates of point B)

### Step 2: Set up the equations using the midpoint formula
From the midpoint formula, we can set up the following equations:

1. For the x-coordinates:
\[ 2 = \frac{11 + x_2}{2} \]

2. For the y-coordinates:
\[ 1 = \frac{5 + y_2}{2} \]

### Step 3: Solve for \( x_2 \)
To find \( x_2 \), we can rearrange the first equation:
\[ 2 = \frac{11 + x_2}{2} \]
Multiply both sides by 2:
\[ 4 = 11 + x_2 \]
Now, isolate \( x_2 \):
\[ x_2 = 4 - 11 = -7 \]

### Step 4: Solve for \( y_2 \)
Now, we solve for \( y_2 \) using the second equation:
\[ 1 = \frac{5 + y_2}{2} \]
Multiply both sides by 2:
\[ 2 = 5 + y_2 \]
Now, isolate \( y_2 \):
\[ y_2 = 2 - 5 = -3 \]

### Step 5: Write the coordinates of point B
Now we have found the coordinates of point B:
\[ (x_2, y_2) = (-7, -3) \]

### Final Answer
The coordinates of point B are \( (-7, -3) \).

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