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If PR = 50, PQ = 3x – 20, and QR = 2x + 10.

Find the value of PQ and QR.


Let's solve the problem using the information provided.

### Given:
- \( PR = 50 \)
- \( PQ = 3x - 20 \)
- \( QR = 2x + 10 \)

Since \( P \), \( Q \), and \( R \) are collinear points and \( Q \) lies between \( P \) and \( R \), we have:
\[
PQ + QR = PR
\]

Substituting the given values:
\[
(3x - 20) + (2x + 10) = 50
\]

Combine like terms:
\[
5x - 10 = 50
\]

Now, solve for \( x \):
\[
5x = 60
\]
\[
x = 12
\]

### Finding \( PQ \) and \( QR \):
1. Substitute \( x = 12 \) into the expression for \( PQ \):
\[
PQ = 3(12) - 20 = 36 - 20 = 16
\]

2. Substitute \( x = 12 \) into the expression for \( QR \):
\[
QR = 2(12) + 10 = 24 + 10 = 34
\]

### Final Values:
- \( PQ = 16 \)
- \( QR = 34 \)

Would you like further details or explanations?

Here are five related questions to expand your understanding:
1. How do you verify if a point lies exactly between two other points on a line?
2. What does the term "collinear" mean in geometry?
3. How would the solution change if \( Q \) were not between \( P \) and \( R \)?
4. Can a similar method be used to find missing segment lengths in non-linear configurations?
5. How can you check if the calculated lengths \( PQ \) and \( QR \) are correct?

**Tip:** Always check your work by substituting your solution back into the original equation to ensure that everything adds up correctly.

Learn how to calculate the lengths of line segments PQ and QR when given the total length PR = 50 and algebraic expressions for PQ and QR. This problem involves solving linear equations and understanding collinear points in geometry.

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