Suppose Q is the midpoint of PR. Use the information to find the missing value. PQ = 6x + 25 and QR = 16 – 3x; Find PR.

Given that \( Q \) is the midpoint of \( PR \), we know that the distances \( PQ \) and \( QR \) are equal. Therefore, we can set up the equation:

\[
PQ = QR
\]

Substituting the given expressions:

\[
6x + 25 = 16 - 3x
\]

Now, let's solve for \( x \):

1. Add \( 3x \) to both sides to get all terms involving \( x \) on one side:

\[
6x + 3x + 25 = 16
\]

\[
9x + 25 = 16
\]

2. Subtract 25 from both sides:

\[
9x = 16 - 25
\]

\[
9x = -9
\]

3. Divide both sides by 9:

\[
x = -1
\]

Now that we have \( x = -1 \), we can substitute this value back into the expressions for \( PQ \) and \( QR \) to find their lengths.

For \( PQ \):

\[
PQ = 6(-1) + 25 = -6 + 25 = 19
\]

For \( QR \):

\[
QR = 16 - 3(-1) = 16 + 3 = 19
\]

Since \( Q \) is the midpoint, \( PR \) is twice the length of either \( PQ \) or \( QR \):

\[
PR = PQ + QR = 19 + 19 = 38
\]

Thus, the length of \( PR \) is \( 38 \).

Would you like further details or have any questions? Here are some related questions:

1. What would \( PR \) be if \( PQ = 4x + 10 \) and \( QR = 8x - 2 \)?
2. How does the concept of a midpoint help in solving problems involving line segments?
3. What happens to the value of \( PR \) if \( x \) changes?
4. How can the midpoint formula be used in coordinate geometry?
5. Can the length of \( PR \) be negative? Why or why not?

**Tip:** Always double-check your algebraic manipulations to ensure accuracy in solving equations.

Learn how to find the length of line segment PR when Q is the midpoint and segments PQ = 6x + 25 and QR = 16 - 3x are given. Detailed algebraic steps and geometric concepts involved in solving the problem are covered.

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