In the given right triangle $\triangle ABC$, the side lengths are as follows:

• $AB = c$ is the hypotenuse,
• $BC = 8$, and
• $AC = 9$.

We can use the Pythagorean theorem to find the length of the hypotenuse $c$. The theorem states:

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

Where $a = 8$ and $b = 9$, so:

$c^2 = 8^2 + 9^2 = 64 + 81 = 145$

Thus,

$c = \sqrt{145}$

Now, let’s approximate $\sqrt{145}$ to determine between which two whole numbers it falls.

$\sqrt{145} \approx 12.041$

Therefore, the length of $c$ lies between 12 and 13.

The correct answer is: 12 and 13.

Would you like any further details or explanations? Here are 5 related questions for further exploration:

1. How does the Pythagorean theorem apply to different types of triangles?
2. What is the geometric meaning of the square root when applied to the Pythagorean theorem?
3. How can we calculate square roots by hand?
4. What other methods can be used to estimate the length of a triangle's side?
5. How does the Pythagorean theorem help in finding distances in coordinate geometry?

Tip: When dealing with right triangles, always check if it's a Pythagorean triple, which are specific sets of whole numbers for the sides of a right triangle. This can save time in calculations.