 Solve for $x.$

[asy] unitsize(1 cm); pair A, B, M, P; A = (-2,0); B = (2,0); M = (0,0); P = (0,4); draw(A--B); draw(A--P--B); draw(P--M); draw(rightanglemark(P,M,B,8)); add(pathticks(A--M, 1, .5, 6, 8)); add(pathticks(B--M, 1, .5, 6, 8)); label("$A$", A, W); label("$B$", B, E); label("$P$", P, N); label("$3x - 17$", (A + P)/2, NW, red); label("$x + 9$", (B + P)/2, NE, red); [/asy]


We are given a geometric figure where $A$, $B$, and $P$ form a triangle, and $M$ is the midpoint of segment $AB$. We also know that there are two distances marked on the diagram: the distance from $A$ to $P$ is $3x - 17$ and the distance from $B$ to $P$ is $x + 9$. Additionally, the line from $P$ to $M$ is perpendicular to segment $AB$, forming a right angle.

Since $M$ is the midpoint of segment $AB$, and the line $PM$ is perpendicular to $AB$, the triangle $APB$ is isosceles. This implies that the distances from $A$ to $P$ and from $B$ to $P$ must be equal. Therefore, we can set the two expressions for the distances equal to each other:

\[
3x - 17 = x + 9.
\]

### Step 1: Solve for $x$
To solve this equation, we first subtract $x$ from both sides:

\[
3x - x - 17 = 9,
\]
\[
2x - 17 = 9.
\]

Next, add 17 to both sides:

\[
2x = 26.
\]

Finally, divide by 2:

\[
x = 13.
\]

### Conclusion:
The value of $x$ is $ \boxed{13} $.

Would you like more details on any part of the solution?

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Here are 5 related questions you might explore:
1. What is the significance of an isosceles triangle in geometric proofs?
2. How do you prove that the perpendicular from the vertex of an isosceles triangle to the base bisects the base?
3. How can you use coordinate geometry to verify the properties of this triangle?
4. What would change if the distances were not equal in this problem?
5. How do geometric principles apply in solving algebraic equations in real-world contexts?

**Tip**: In isosceles triangles, recognizing symmetry helps reduce the complexity of problems and identify equal lengths or angles quickly.

Solve for $x.$ [asy] unitsize(1 cm); pair A, B, M, P; A = (-2,0); B = (2,0); M = (0,0); P = (0,4); draw(A--B); draw(A--P--B); draw(P--M); draw(rightanglemark(P,M,B,8)); add(pathticks(A--M, 1, .5, 6, 8)); add(pathticks(B--M, 1, .5, 6, 8)); label("$A$", A, W); label("$B$", B, E); label("$P$", P, N); label("$3x - 17$", (A + P)/2, NW, red); label("$x + 9$", (B + P)/2, NE, red); [/asy]

Learn how to solve for x in a geometric problem involving an isosceles triangle. The problem features algebraic equations, geometric properties of isosceles triangles, and key theorems like the Isosceles Triangle Theorem. Suitable for Grades 8-10.

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