Question

Line n is a perpendicular bisector of line segment T V. It intersects line segment T V at point R. Line n also contains points Q and S. Line segment R V is 3 x + 2. Line segment Q V is 4 x + 1. Line segment T S is 9 x minus 4.
The length of TR is 17 units. What are the lengths of SV and QT?

SV =
units

QT =
units

Answer 1

Given the information provided, we can start solving the problem step by step:

1. **Understanding the setup:**
- Line segment \( TV \) is bisected by line \( n \) at point \( R \), which means \( TR = RV \).
- Since \( TR = 17 \) units, then \( RV = 17 \) units as well. This gives us the total length of segment \( TV \):
\[
TV = TR + RV = 17 + 17 = 34 \text{ units.}
\]

2. **Expression for segment lengths:**
- From the problem, we know that:
- \( RV = 3x + 2 \)
- \( QV = 4x + 1 \)
- \( TS = 9x - 4 \)
- Since \( RV = 17 \), we can set up the equation:
\[
3x + 2 = 17.
\]

3. **Solving for \( x \):**
- Subtract 2 from both sides:
\[
3x = 15.
\]
- Divide both sides by 3:
\[
x = 5.
\]

4. **Calculating the lengths of \( QV \) and \( TS \):**
- Now we can find \( QV \):
\[
QV = 4x + 1 = 4(5) + 1 = 20 + 1 = 21 \text{ units.}
\]
- Next, we calculate \( TS \):
\[
TS = 9x - 4 = 9(5) - 4 = 45 - 4 = 41 \text{ units.}
\]

5. **Finding lengths of \( SV \) and \( QT \):**
- Since line \( n \) is the perpendicular bisector, we know:
- \( ST = TS \), which implies \( SV = RV = 17 \) (since \( SV = ST \)).
- For \( QT \), since \( RV + QT = QV\), we find:
\[
QT = QV - RV = 21 - 17 = 4 \text{ units.}
\]

Now we summarize the results:
- The length of \( SV \) is 17 units.
- The length of \( QT \) is 4 units.

Thus, we have:
- \( SV = 17 \) units
- \( QT = 4 \) units