Question

what is the length of $overline{ps}$, when $r$ is the mid - point of $overline{ps}$?
$9x - 6$ $2(x + 4)$

Explanation:

Step1: Set up equation using mid - point property

Since $R$ is the mid - point of $\overline{PS}$, then $PR = RS$. So, $9x - 6=2(x + 4)$.

Step2: Expand the right - hand side

Expand $2(x + 4)$ to get $2x+8$. The equation becomes $9x - 6=2x + 8$.

Step3: Solve for $x$

Subtract $2x$ from both sides: $9x-2x - 6=2x-2x + 8$, which simplifies to $7x-6 = 8$. Then add 6 to both sides: $7x-6 + 6=8 + 6$, so $7x=14$. Divide both sides by 7: $x=\frac{14}{7}=2$.

Step4: Find the length of $PR$ and $RS$

Substitute $x = 2$ into the expression for $PR$: $PR=9x - 6=9\times2-6=18 - 6=12$. Substitute $x = 2$ into the expression for $RS$: $RS=2(x + 4)=2\times(2 + 4)=2\times6 = 12$.

Step5: Calculate the length of $PS$

Since $PS=PR + RS$, and $PR = RS = 12$, then $PS=12+12 = 24$.

Answer:

24