Let's solve these two problems step-by-step:

Problem 27

Given:

• $P$ is the circumcenter of $\triangle ABC$.
• $AD = 8x + 3$, $DB = 17x - 15$, and $DP = 12$.

Since $P$ is the circumcenter, $P$ is equidistant from all vertices of $\triangle ABC$. This implies that $DP = PB$. Thus:

1. Set $DP = PB$.
2. Substitute $DP = 12$ to find $PB = 12$.

Therefore, $PB = 12$.

Problem 29

Given:

• $J$ is the incenter of $\triangle DEF$.
• $m \angle DEJ = 6x + 1$
• $m \angle JEF = 10x - 23$
• We need to find $m \angle FDJ$.

Since $J$ is the incenter, the angles $\angle DEJ$ and $\angle JEF$ are parts of $\angle DEF$, and these angles form a linear pair with the remaining portion of the angle at $J$. We also know the full angle around $J$ adds up to 90°.

To solve this:

1. Set up an equation for the sum of these two expressions: $(6x + 1) + (10x - 23) = 25$.
2. Solve for $x$ and use it to find the individual measures of the angles.

Would you like further details on either problem, or have any questions?

Related Questions

1. What is the role of a circumcenter in a triangle?
2. How is the incenter used to find angle measures within a triangle?
3. Can you explain why $DP = PB$ in circumcenter problems?
4. How do we solve linear equations involving geometric measures?
5. What are other methods to verify angle measures around a triangle’s incenter?

Tip

Always remember that the circumcenter equidistantly connects to the triangle's vertices, while the incenter touches each side at the midpoint of the angles formed at the triangle's vertices.