Question

1. if ro = 5x and rq = 12x - 20, then find the value of x. then find rq and ro.
2. in exercises 51 and 52, identify the segment bisector of st. then find st.

8.

Explanation:

Step1: Set up the equation

Since we assume there is some relationship (perhaps related to a bisector - not fully clear from the problem description but from the context of line - segments), we set up the equation based on the given lengths. If we assume $RO = RQ$, then $5x=12x - 20$.

Step2: Rearrange the equation

Subtract $5x$ from both sides: $0 = 12x-5x - 20$, which simplifies to $0 = 7x - 20$.

Step3: Solve for $x$

Add 20 to both sides: $20 = 7x$. Then divide both sides by 7, so $x=\frac{20}{7}$.

Step4: Find the length of $RO$

Substitute $x = \frac{20}{7}$ into the expression for $RO$. $RO = 5x=5\times\frac{20}{7}=\frac{100}{7}$.

Step5: Find the length of $RQ$

Substitute $x=\frac{20}{7}$ into the expression for $RQ$. $RQ = 12x - 20=12\times\frac{20}{7}-20=\frac{240}{7}-\frac{140}{7}=\frac{100}{7}$.

Answer:

$x=\frac{20}{7}$, $RO=\frac{100}{7}$, $RQ=\frac{100}{7}$