The problem describes a right triangle with legs of 24 cm and 21 cm. We need to find the length of the hypotenuse, which is expressed in the form $3\sqrt{d}$, where $d$ is an integer. The task is to determine the value of $d$.

Step-by-step solution:

1. Apply the Pythagorean Theorem:
   The Pythagorean theorem for a right triangle is:
   $c^2 = a^2 + b^2$ where $c$ is the hypotenuse and $a$ and $b$ are the legs.

2. Substitute the given values:
   $c^2 = 24^2 + 21^2$ Let's calculate the squares of 24 and 21:
   $24^2 = 576, \quad 21^2 = 441$ Adding them together:
   $c^2 = 576 + 441 = 1017$

3. Solve for $c$:
   We know $c^2 = 1017$, so:
   $c = \sqrt{1017}$

4. Express the hypotenuse in the form $3\sqrt{d}$:
   We are told that the hypotenuse can be written as $3\sqrt{d}$, meaning:
   $\sqrt{1017} = 3\sqrt{d}$ Squaring both sides:
   $1017 = 9d$ Solving for $d$:
   $d = \frac{1017}{9} = 113$

Thus, the value of $d$ is 113.

Do you want more details or have any other questions?

Here are 5 related questions to explore further:

1. How would the value of $d$ change if the legs of the triangle were different?
2. Can the Pythagorean theorem be applied to non-right triangles? Why or why not?
3. What are some real-life applications of the Pythagorean theorem?
4. How would you calculate the area of this triangle?
5. Can a right triangle have sides that are non-integer values?

Tip: Always simplify square roots and ratios to make calculations easier when dealing with geometric figures!