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.

Question

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