In ΔXYZ, XY and XZ are 20 cm and 25 cm respectively, XP is angle bisector of ∠YXZ, YP = $a$ cm and PZ = (a+3) cm. If I is the incenter of the triangle, then find the ratio of XI : IP.

Step 1 (Apply Angle Bisector Theorem on Outer Triangle): In $△XYZ$, $XP$ is the interior angle bisector of $\angle YXZ$. By the Angle Bisector Theorem:

We are given $XY=20$, $XZ=25$, $YP=a$, and $PZ=a+3$:

So $YP=12$ cm and $PZ=15$ cm.

Step 2 (Find the Incenter Ratio): The incenter $I$ is the intersection of all three angle bisectors of $△XYZ$. In $△XYP$, the bisector of $\angle XYP$ is $YI$, which intersects the bisector $XP$ at the incenter $I$.
Applying the Angle Bisector Theorem on $△XYP$ with bisector $YI$:

Substitute $XY=20$ and $YP=12$:

Step 3 (Conclusion): The ratio $XI:IP$ is exactly 5/3 (or 5:3).