We are tasked with finding the probability that triangle $\triangle ABX$, where point $X$ is randomly chosen inside triangle $\triangle ABC$, has an area greater than 25.

Step-by-step Solution:

1. Find the area of triangle $\triangle ABC$: Since triangle $\triangle ABC$ is a right triangle, we can use the formula for the area of a triangle: $\text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base $AB = 5$ and the height $AC = 12$. So, the area is: $\text{Area of } \triangle ABC = \frac{1}{2} \times 5 \times 12 = 30$

2. Understanding the problem geometrically: The point $X$ is randomly chosen from within the triangle $\triangle ABC$. When we select point $X$, the triangle $\triangle ABX$ will have an area that is some fraction of the area of $\triangle ABC$.

   Since $\triangle ABC$ has an area of 30, we want to find the probability that the area of triangle $\triangle ABX$ is greater than 25.

3. Set up the proportion: The area of triangle $\triangle ABX$ can be written as a fraction of the area of triangle $\triangle ABC$. Let $\lambda$ be the fraction such that the area of $\triangle ABX$ is $\lambda \times 30$. We need to solve for when this area is greater than 25: $\lambda \times 30 > 25$ Solving for $\lambda$: $\lambda > \frac{25}{30} = \frac{5}{6}$

4. Determine the probability: Since $X$ is randomly chosen from within $\triangle ABC$, the area ratio follows a uniform distribution. The probability that the area of $\triangle ABX$ is greater than 25 corresponds to the fraction of $\lambda$ values greater than $\frac{5}{6}$.

   The possible values of $\lambda$ range from 0 to 1. Thus, the probability that $\lambda > \frac{5}{6}$ is: $P(\lambda > \frac{5}{6}) = 1 - \frac{5}{6} = \frac{1}{6}$

Final Answer:

The probability that the area of triangle $\triangle ABX$ is greater than 25 is $\boxed{\frac{1}{6}}$.

Would you like further details or have any other questions?

Here are 5 related questions you may find interesting:

1. How does the random selection of point $X$ inside $\triangle ABC$ influence the area of $\triangle ABX$?
2. Can you calculate the probability that $\triangle ABX$ has an area less than 15?
3. How would the solution change if triangle $ABC$ was not a right triangle?
4. What if the area of triangle $ABX$ had to be exactly 25? What would the probability be then?
5. How can we generalize this problem to triangles with different base and height dimensions?

Tip: In geometric probability, the area ratios often give the simplest way to calculate probabilities when dealing with random points within a region.