Question

If \( M\left(\frac{p}{3}, 4\right) \) is the midpoint of the line segment joining \( A(-6, 5) \) and \( B(-4, 3) \), then \( p = ? \)

• A. \( -10 \)
• B. \( -8 \)
• C. \( -9 \)
• D. \( -15 \)

Hint:

The midpoint formula is essential for coordinate geometry problems involving line segments. Remember to average the x-coordinates and the y-coordinates separately.

Answer 1

The Correct Option is D

Step 1: Recall the midpoint formula.
The midpoint \( M(x_m, y_m) \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \] Step 2: Apply the midpoint formula to the given points.
We are given \( A(-6, 5) \), \( B(-4, 3) \), and the midpoint \( M\left(\frac{p}{3}, 4\right) \). Using the midpoint formula for the x-coordinate: \[ \frac{p}{3} = \frac{-6 + (-4)}{2} \] Using the midpoint formula for the y-coordinate: \[ 4 = \frac{5 + 3}{2} \] Step 3: Solve for \( p \) using the x-coordinate equation. \[ \frac{p}{3} = \frac{-6 - 4}{2} \] \[ \frac{p}{3} = \frac{-10}{2} \] \[ \frac{p}{3} = -5 \] Multiply both sides by 3 to find \( p \): \[ p = -5 \times 3 \] \[ p = -15 \] Step 4: Verify the y-coordinate.
Let's check if the y-coordinate of the midpoint matches: \[ \frac{5 + 3}{2} = \frac{8}{2} = 4 \] The y-coordinate matches the given midpoint. Thus, the value of \( p \) is \( -15 \).