We are given a circle with center $P$. $AY = 24$ yd. We want to find the lengths of $AP$ and $XB$.

1. Problem Description

We are given a circle with center $P$. $AY = 24$ yd. We want to find the lengths of $AP$ and $XB$.

2. Solution Steps

Since $P$ is the center of the circle, $AP$ and $PY$ are radii of the circle. Therefore, $AP = PY$. We also know that $AY = 24$ yd. We can write the equation:

$AP + PY = AY$

Since $AP = PY$, we can substitute $AP$ for $PY$ in the equation:

$AP + AP = 24$

$2AP = 24$

$AP = \frac{24}{2}$

$AP = 12$

So, $AP = 12$ yd.

Similarly, $PX$ and $PB$ are radii of the circle, so $PX = PB = AP$. $XB = PX + PB$. Since $PX = PB$, $XB=PB + PX$ becomes $XB = PX +PX$ or $XB= PB + PB$.

$XB= 2PX$. We know that $AP = PX$, so $PX = 12$.

Then, $XB = 2(12) = 24$.

So, $XB = 24$ yd.

3. Final Answer

$AP = 12$ yd

$XB = 24$ yd