The mid-point of (3p,4) and (-2,2q) is (2,6) The value of (p + q) is:

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) given that the midpoint of the points \( (3p, 4) \) and \( (-2, 2q) \) is \( (2, 6) \).

### Step 1: Set up the midpoint formula
The midpoint \( M \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
In our case, the points are \( (3p, 4) \) and \( (-2, 2q) \). Therefore, we can write:
\[ M = \left( \frac{3p + (-2)}{2}, \frac{4 + 2q}{2} \right) \]
We know that this midpoint is equal to \( (2, 6) \).

### Step 2: Set up equations from the midpoint
From the x-coordinates:
\[ \frac{3p - 2}{2} = 2 \]
From the y-coordinates:
\[ \frac{4 + 2q}{2} = 6 \]

### Step 3: Solve the x-coordinate equation
Multiply both sides of the x-coordinate equation by 2:
\[ 3p - 2 = 4 \]
Now, add 2 to both sides:
\[ 3p = 6 \]
Divide by 3:
\[ p = 2 \]

### Step 4: Solve the y-coordinate equation
Multiply both sides of the y-coordinate equation by 2:
\[ 4 + 2q = 12 \]
Subtract 4 from both sides:
\[ 2q = 8 \]
Divide by 2:
\[ q = 4 \]

### Step 5: Find \( p + q \)
Now that we have \( p = 2 \) and \( q = 4 \), we can find \( p + q \):
\[ p + q = 2 + 4 = 6 \]

### Final Answer
The value of \( p + q \) is \( 6 \).

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