P(x, 4) and Q( 10, 8) are two points joined by a straight line in a plane. If the midpoint of the line is (9, 6), find the value of x.

1. Analyze the question

We are given two points, $P(x, 4)$ and $Q(10, 8)$, and their midpoint $M(9, 6)$. We need to find the unknown coordinate $x$.

2. Explain the concept

In coordinate geometry, the midpoint $M(x_m, y_m)$ of a line segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the midpoint formula:

Since we are looking for $x$, we only need to focus on the formula for the x-coordinate.

3. Show the solution

Given:

• $P = (x_1, y_1) = (x, 4)$
• $Q = (x_2, y_2) = (10, 8)$
• Midpoint $M = (x_m, y_m) = (9, 6)$

Using the formula for the x-coordinate of the midpoint:

Substitute the known values into the equation:

Now, solve for $x$:

1. Multiply both sides by 2 to clear the fraction: $$9 \times 2 = x + 10$$ $$18 = x + 10$$

2. Subtract 10 from both sides: $$x = 18 - 10$$ $$x = 8$$

(Self-check: If we check the y-coordinates, $\frac{4 + 8}{2} = \frac{12}{2} = 6$. This matches our midpoint y-coordinate, confirming our calculation is correct.)

4. Identify the answer

The value of $x$ is 8, which corresponds to option C.

5. Tip

When dealing with midpoint questions, always remember that the midpoint is simply the average of the x-coordinates and the average of the y-coordinates. If you ever forget the formula, just think: "Add them up and divide by two!"