Question

1. \\( \overline { v x } \\) is an angle bisector. write and solve an equation

to find \\( r \\). then find \\( u x, x w \\), and \\( u w \\).

\\( r = \\) \\( u x = \\) \\( x w = \\) \\( u w = \\)

Explanation:

Step1: Use the Angle Bisector Theorem

Since \( \overline{VX} \) is an angle bisector and \( \overline{VU}=\overline{VW} \), then \( UX = XW \). So, \( 5r + 9=8r \).

Step2: Solve for \( r \)

Subtract \( 5r \) from both sides: \( 9 = 8r-5r \), which simplifies to \( 9 = 3r \). Then divide both sides by 3: \( r=\frac{9}{3}=3 \).

Step3: Find \( UX \)

Substitute \( r = 3 \) into \( UX=5r + 9 \). So, \( UX=5\times3+9=15 + 9=24 \).

Step4: Find \( XW \)

Substitute \( r = 3 \) into \( XW = 8r \). So, \( XW=8\times3=24 \).

Step5: Find \( UW \)

\( UW=UX + XW \). Since \( UX = 24 \) and \( XW = 24 \), then \( UW=24+24=48 \).

Answer:

\( r = 3 \), \( UX = 24 \), \( XW = 24 \), \( UW = 48 \)