Question

Point \( P \left( \frac{a}{8}, 4 \right) \) is the mid-point of the line segment joining the points \( A(-4, 2) \) and \( B(5, 6) \). The value of \( a \) will be:

• A. -8
• B. -4
• C. 2
• D. 4

Hint:

The midpoint of two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is found using the formula \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Answer 1

The Correct Option is B

The midpoint formula for the coordinates of two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Step 1: Apply the midpoint formula.
We are given: - \( A(-4, 2) \) - \( B(5, 6) \) - Midpoint \( P \left( \frac{a}{8}, 4 \right) \) Using the midpoint formula for the x-coordinates: \[ \frac{x_1 + x_2}{2} = \frac{-4 + 5}{2} = \frac{1}{2} \] For the y-coordinates: \[ \frac{y_1 + y_2}{2} = \frac{2 + 6}{2} = 4 \] We already know that the y-coordinate of the midpoint is 4, which matches the given value for point \( P \).
Step 2: Solve for \( a \).
From the x-coordinate of the midpoint: \[ \frac{a}{8} = \frac{1}{2} \] Multiplying both sides by 8: \[ a = 4 \]
Step 3: Conclusion.
Therefore, the value of \( a \) is \( -4 \).