We are given an equilateral triangle where the height is $k \, \text{cm}$, and the right triangle formed by splitting the equilateral triangle has one leg of $9 \, \text{cm}$ and the hypotenuse $p \, \text{cm}$. We need to express $p$ in terms of $k$.

Step-by-Step Explanation:

1. Identify the structure of the triangle:

   • In an equilateral triangle, the height divides the triangle into two right-angled triangles.
   • One leg of this right triangle is the height $k$.
   • The other leg is half of the base of the equilateral triangle, which is given as $9 \, \text{cm}$.
   • The hypotenuse of this right triangle is the side length $p$ of the equilateral triangle.

2. Apply the Pythagorean theorem: The Pythagorean theorem states that for any right triangle, the square of the hypotenuse $p$ is the sum of the squares of the other two sides (the legs). Here, the two legs are $k$ (the height) and $9 \, \text{cm}$ (half of the base).

   So, using the Pythagorean theorem: $p^2 = k^2 + 9^2$ Simplifying this: $p^2 = k^2 + 81$

3. Solve for $p$: To find $p$, take the square root of both sides: $p = \sqrt{k^2 + 81}$

Thus, the length of one side $p$ of the equilateral triangle, in terms of $k$, is: $p = \sqrt{k^2 + 81}$

Would you like more details on any part of the solution? Here are five related questions you could explore:

1. How does the height of an equilateral triangle relate to its side length?
2. How do we derive the formula for the height of an equilateral triangle in terms of its side length?
3. What happens to the side length $p$ if the height $k$ increases?
4. How can the Pythagorean theorem be used in other types of triangles?
5. What are the properties of an equilateral triangle, and how do they help in problem-solving?

Tip: Always remember to check the relationships between different elements of a triangle, such as angles and sides, as they often lead to quick solutions using geometric principles like the Pythagorean theorem.