Question

$\overline{pq}$ is bisected at $r$. if $pq = 8x + 24$ and $rq = 8x - 12$, what is the length of $\overline{rq}$?

Explanation:

Step1: Recall the definition of a bisector

Since \( \overline{PQ} \) is bisected at \( R \), we know that \( PQ = 2 \times RQ \). This is because a bisector divides a segment into two equal parts, so the length of the whole segment \( PQ \) is twice the length of each of the smaller segments \( PR \) and \( RQ \), and since \( PR = RQ \) (because \( R \) is the midpoint), \( PQ = 2RQ \).

Step2: Set up the equation

We are given that \( PQ = 8x + 24 \) and \( RQ = 8x - 12 \). Using the relationship from Step 1, we can set up the equation:

$$ 8x + 24 = 2(8x - 12) $$

Step3: Solve the equation for \( x \)

First, expand the right - hand side of the equation:

$$ 8x + 24 = 16x-24 $$

Then, subtract \( 8x \) from both sides:

$$ 24=8x - 24 $$

Next, add 24 to both sides:

$$ 48 = 8x $$

Finally, divide both sides by 8:

$$ x = 6 $$

Step4: Find the length of \( \overline{RQ} \)

Now that we know \( x = 6 \), we substitute \( x = 6 \) into the expression for \( RQ \), which is \( RQ=8x - 12 \).

$$ RQ=8\times6-12 $$

$$ RQ = 48-12 $$

$$ RQ=36 $$

Answer:

36