To solve for $x$ in this right triangle using the Pythagorean theorem, we can apply the formula:

$a^2 + b^2 = c^2$

where:

• $a = 24 \, \text{cm}$,
• $b = 31 \, \text{cm}$,
• $c = x$ (the hypotenuse).

Thus:

$x^2 = 24^2 + 31^2$

Calculating each square:

$24^2 = 576$ $31^2 = 961$

Adding these results:

$x^2 = 576 + 961 = 1537$

Now, take the square root to find $x$:

$x = \sqrt{1537} \approx 39.2 \, \text{cm}$

So, the value of $x$ is approximately 39.2 cm.

Would you like more details, or do you have any questions?

Here are five related questions that expand on this concept:

1. How would you solve if the hypotenuse were given and one leg was unknown?
2. How does the Pythagorean theorem apply to non-Euclidean geometries?
3. How can we confirm if a triangle with three given side lengths is a right triangle?
4. What are the practical applications of the Pythagorean theorem in construction?
5. How can you derive the Pythagorean theorem from Euclidean geometry principles?

Tip: Remember that the Pythagorean theorem only applies to right-angled triangles.