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We are given that \( P \) is the incenter of triangle \( RQS \). The incenter is the point where the angle bisectors of a triangle intersect and is equidistant from all three sides. This means that \( PJ = PK = PL \) because they are the perpendicular distances from the incenter to the sides of the triangle.

We are also given the following equations:
- \( PJ = 4x - 8 \)
- \( PL = x + 7 \)

Since \( P \) is the incenter, \( PJ = PL = PK \). Therefore, we can set \( PJ \) equal to \( PL \) and solve for \( x \).

### Step 1: Set \( PJ = PL \)
\[
4x - 8 = x + 7
\]

### Step 2: Solve for \( x \)
\[
4x - x = 7 + 8
\]
\[
3x = 15
\]
\[
x = 5
\]

### Step 3: Find \( PJ \) and \( PK \)
Substitute \( x = 5 \) into the expression for \( PJ \):
\[
PJ = 4(5) - 8 = 20 - 8 = 12
\]

Since \( PJ = PK \), we have \( PK = 12 \).

Thus, the answer is \( PK = 12 \), which corresponds to option **a**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the geometric significance of the incenter in a triangle?
2. How do you find the incenter of a triangle given the coordinates of its vertices?
3. What is the difference between the incenter and the circumcenter of a triangle?
4. How do you construct the angle bisectors of a triangle?
5. How do perpendicular distances from the incenter to the triangle's sides relate to the area of the triangle?

**Tip**: The incenter is always inside the triangle, regardless of the triangle's shape (acute, obtuse, or right).

P is the incenter, find the indicated measure. PJ = 4x - 8, PL = x + 7. Find PK.

Learn how to find the measure of PK when P is the incenter of triangle RQS, given PJ = 4x - 8 and PL = x + 7. This problem covers key geometry concepts, including the Incenter Theorem and solving algebraic equations. Suitable for students in Grades 9-10.

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