Two chords, $AB$ and $CD,$ meet inside a circle at $P.$  If $AP = 3$ and $CP = 8,$ then what is $\frac{BP}{DP}$?
By the Power of a Point formula, we know that $AP \cdot BP = CP \cdot DP.$ Substituting, we have $3 \cdot BP = 8 \cdot DP.$ Then, we have $\frac{BP}{DP} = \boxed{\frac{8}{3}}.$
\frac{8}{3}
Level 3
Geometry
Problem: Two chords, $AB$ and $CD,$ meet inside a circle at $P.$  If $AP = 3$ and $CP = 8,$ then what is $\frac{BP}{DP}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the ratio $\frac{BP}{DP}$, we can use the property of intersecting chords in a circle. This property states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. 

Let $BP = x$ and $DP = y$. According to the intersecting chords theorem, we have:

1. For chord $AB$: The segments are $AP$ and $BP$, so the product is $AP \cdot BP = 3 \cdot x$.
2. For chord $CD$: The segments are $CP$ and $DP$, so the product is $CP \cdot DP = 8 \cdot y$.

Setting these two products equal gives us the equation:

3. $3x = 8y$.

We need to find the ratio $\frac{BP}{DP}$, which is $\frac{x}{y}$. To do this, we can rearrange the equation:

4. From $3x = 8y$, we can express $x$ in terms of $y$: $x = \frac{8}{3}y$.

5. Now, substituting this expression into the ratio $\frac{x}{y}$ gives us:

6. $\frac{x}{y} = \frac{\frac{8}{3}y}{y} = \frac{8}{3}$.

Thus, the final answer is $\frac{8}{3}$.<|END_OF_LATENT|> The final answer is $\frac{8}{3}$.