In the given circle, $S$ is the center. $JK = 4x - 2$, $LM = 10$, $SN = 8$, and $SP = 8$. $SN$ and $SP$ are perpendicular to chords $JK$ and $LM$, respectively. We need to find the values of $x$ and $PL$.

1. Problem Description

In the given circle, $S$ is the center. $JK = 4x - 2$, $LM = 10$, $SN = 8$, and $SP = 8$. $SN$ and $SP$ are perpendicular to chords $JK$ and $LM$, respectively. We need to find the values of $x$ and $PL$.

2. Solution Steps

Since $S$ is the center of the circle and $SN$ and $SP$ are perpendicular to chords $JK$ and $LM$ respectively, $SN$ bisects $JK$ and $SP$ bisects $LM$. Therefore, $JN = \frac{1}{2}JK$ and $LP = \frac{1}{2}LM$. Also, $SN = SP = 8$, which means the chords $JK$ and $LM$ are equidistant from the center. Hence, they must have equal lengths. Therefore, $JK = LM$.

So, we have $4x - 2 = 10$.

Now, we solve for $x$:

$4x - 2 = 10$

$4x = 10 + 2$

$4x = 12$

$x = \frac{12}{4}$

$x = 3$

Now, we find $PL$. Since $LP = \frac{1}{2}LM$ and $LM = 10$,

$LP = \frac{1}{2} (10)$

$LP = 5$

3. Final Answer

$x = 3$

$PL = 5$