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.

The midpoint of a line segment joining two points is the average of the x-coordinates and the y-coordinates of the endpoints. Given two endpoints \( P(x, 4) \) and \( Q(10, 8) \), and their midpoint \( M(9, 6) \), we can use the midpoint formula to find the value of \( x \).

The formula for the midpoint \( M(x_m, y_m) \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\( x_m = \frac{x_1 + x_2}{2} \)

\( y_m = \frac{y_1 + y_2}{2} \)

Given that \( M(9, 6) \) is the midpoint, we know:

\(\frac{x + 10}{2} = 9\)

\(\frac{4 + 8}{2} = 6\)

Let’s solve for \( x \):

Multiply both sides of the x-coordinate equation by 2 to eliminate the fraction:

\( x + 10 = 18 \)

Subtract 10 from both sides:

\( x = 18 - 10 \)

Thus:

x = 8

Therefore, the value of \( x \) is 8. This makes sure the midpoint of the line segment \( P(x, 4) \) and \( Q(10, 8) \) is indeed \( M(9, 6) \). The answer aligns with the given options.