Suppose Q is the midpoint of PR. Use the information to find the missing value. 13.  PQ = 3x + 14 and QR = 7x – 10; Find x. 14.  PQ = 2x + 1 and QR = 5x – 44; Find PQ. 15.  PQ = 6x + 25 and QR = 16 – 3x; Find PR. 16.  PR = 9x – 31 and QR = 43; Find x. 

Let's solve each problem step by step.

### Problem 13: Finding \( x \) when \( Q \) is the midpoint of \( PR \).

Given:
- \( PQ = 3x + 14 \)
- \( QR = 7x - 10 \)

Since \( Q \) is the midpoint of \( PR \), the lengths of \( PQ \) and \( QR \) must be equal:
\[
3x + 14 = 7x - 10
\]

Solve for \( x \):

1. Subtract \( 3x \) from both sides:
   \[
   14 = 4x - 10
   \]
2. Add 10 to both sides:
   \[
   24 = 4x
   \]
3. Divide both sides by 4:
   \[
   x = 6
   \]

So, the value of \( x \) is \( 6 \).

### Problem 14: Finding \( PQ \) when \( Q \) is the midpoint of \( PR \).

Given:
- \( PQ = 2x + 1 \)
- \( QR = 5x - 44 \)

Since \( Q \) is the midpoint of \( PR \), \( PQ = QR \):
\[
2x + 1 = 5x - 44
\]

Solve for \( x \):

1. Subtract \( 2x \) from both sides:
   \[
   1 = 3x - 44
   \]
2. Add 44 to both sides:
   \[
   45 = 3x
   \]
3. Divide both sides by 3:
   \[
   x = 15
   \]

Now, substitute \( x = 15 \) back into the equation for \( PQ \):

\[
PQ = 2(15) + 1 = 30 + 1 = 31
\]

So, the length of \( PQ \) is \( 31 \).

### Problem 15: Finding \( PR \) when \( Q \) is the midpoint of \( PR \).

Given:
- \( PQ = 6x + 25 \)
- \( QR = 16 - 3x \)

Since \( Q \) is the midpoint of \( PR \), \( PQ = QR \):
\[
6x + 25 = 16 - 3x
\]

Solve for \( x \):

1. Add \( 3x \) to both sides:
   \[
   9x + 25 = 16
   \]
2. Subtract 25 from both sides:
   \[
   9x = -9
   \]
3. Divide both sides by 9:
   \[
   x = -1
   \]

Now, find the length of \( PR \):

\( PR = PQ + QR \)

First, substitute \( x = -1 \) into the equations for \( PQ \) and \( QR \):

\[
PQ = 6(-1) + 25 = -6 + 25 = 19
\]
\[
QR = 16 - 3(-1) = 16 + 3 = 19
\]

So, \( PR = 19 + 19 = 38 \).

### Problem 16: Finding \( x \) when \( PR = 9x - 31 \) and \( QR = 43 \).

Given:
- \( PR = 9x - 31 \)
- \( QR = 43 \)

Since \( Q \) is the midpoint of \( PR \), \( QR \) is half of \( PR \). So:
\[
QR = \frac{PR}{2}
\]

Substitute the given values:
\[
43 = \frac{9x - 31}{2}
\]

Solve for \( x \):

1. Multiply both sides by 2:
   \[
   86 = 9x - 31
   \]
2. Add 31 to both sides:
   \[
   117 = 9x
   \]
3. Divide both sides by 9:
   \[
   x = 13
   \]

So, the value of \( x \) is \( 13 \).

---

### Summary of Answers:

1. **Problem 13**: \( x = 6 \)
2. **Problem 14**: \( PQ = 31 \)
3. **Problem 15**: \( PR = 38 \)
4. **Problem 16**: \( x = 13 \)

Would you like a detailed explanation of any step? Here are some related questions:

1. How do you find the length of a segment when given an expression for it?
2. What is the significance of the midpoint in these types of problems?
3. How can you check if your solution for \( x \) is correct?
4. What is the relationship between the lengths of segments and the midpoint?
5. Can you solve problems involving both midpoints and segment addition?

**Tip:** Always verify your solutions by substituting the found value of \( x \) back into the original expressions to ensure consistency.

Explore how to solve midpoint problems in geometry with detailed step-by-step solutions. These problems involve using the midpoint formula and linear equations to find unknown values. Suitable for Grades 7-9 students studying geometry and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation