Two points A (-3, b) and B (1, b + 4) and the coordinates of the middle point of AB are (-1,1). The value of b is

Options

A. `1`
B. `-1`
C. 2
D. 0

Answer

To find the value of \( b \) given the points \( A(-3, b) \) and \( B(1, b + 4) \), and the midpoint of \( AB \) being \( (-1, 1) \), we can follow these steps:

### Step 1: Write down the midpoint formula
The midpoint \( M \) of two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 2: Substitute the coordinates of points A and B into the midpoint formula
Here, \( A(-3, b) \) and \( B(1, b + 4) \). Thus, we can substitute:
\[ M = \left( \frac{-3 + 1}{2}, \frac{b + (b + 4)}{2} \right) \]

### Step 3: Simplify the expressions
Calculating the x-coordinate:
\[ \frac{-3 + 1}{2} = \frac{-2}{2} = -1 \]
Calculating the y-coordinate:
\[ \frac{b + (b + 4)}{2} = \frac{2b + 4}{2} = b + 2 \]

### Step 4: Set the midpoint equal to the given midpoint
We know that the midpoint is \( (-1, 1) \). Therefore, we can set up the following equations:
1. For the x-coordinate:
\[ -1 = -1 \quad \text{(This is true)} \]
2. For the y-coordinate:
\[ b + 2 = 1 \]

### Step 5: Solve for \( b \)
To find \( b \), we solve the equation:
\[ b + 2 = 1 \]
Subtract 2 from both sides:
\[ b = 1 - 2 \]
\[ b = -1 \]

### Final Answer
The value of \( b \) is \( -1 \).

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