If `P((a)/(3),4)` is the mid - point of the line segment joining the points `Q(-6,5)` and `R(-2,3)`, then the value of `a` is

Options

A. `-4`
B. `-12`
C. `12`
D. `-6`

Explanation

To find the value of \( a \) given that \( P\left(\frac{a}{3}, 4\right) \) is the midpoint of the line segment joining the points \( Q(-6, 5) \) and \( R(-2, 3) \), we can follow these steps:

### Step 1: Use the Midpoint Formula
The midpoint \( P \) of a line segment joining two points \( Q(x_1, y_1) \) and \( R(x_2, y_2) \) is given by the formula:
\[ P\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

### Step 2: Identify the Coordinates
Here, the coordinates of points \( Q \) and \( R \) are:
- \( Q(-6, 5) \) where \( x_1 = -6 \) and \( y_1 = 5 \)
- \( R(-2, 3) \) where \( x_2 = -2 \) and \( y_2 = 3 \)

### Step 3: Calculate the Midpoint Coordinates
Using the midpoint formula, we can calculate the x-coordinate and y-coordinate of point \( P \):
- For the x-coordinate:
\[ \frac{x_1 + x_2}{2} = \frac{-6 + (-2)}{2} = \frac{-8}{2} = -4 \]
- For the y-coordinate:
\[ \frac{y_1 + y_2}{2} = \frac{5 + 3}{2} = \frac{8}{2} = 4 \]

### Step 4: Set Up the Equation for x-coordinate
Since we know that the x-coordinate of \( P \) is \( \frac{a}{3} \), we can set up the equation:
\[ \frac{a}{3} = -4 \]

### Step 5: Solve for \( a \)
To find \( a \), multiply both sides of the equation by 3:
\[ a = -4 \times 3 = -12 \]

### Conclusion
Thus, the value of \( a \) is \( -12 \).

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