To determine the possible range for the length of $x$, we are dealing with a right triangle where the two sides are 8 cm and 6 cm, and the right angle is between them.

In a right triangle, the length of the third side $x$ can be determined using the Pythagorean theorem. For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the theorem states:

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

In your case, since the two sides with the right angle are 8 cm and 6 cm, we need to consider both possible situations for $x$ (as the hypotenuse or as one of the legs). Here’s how we analyze each case:

1. $x$ as the Hypotenuse

If $x$ is the hypotenuse, we use the Pythagorean theorem to calculate it:

$x^2 = 8^2 + 6^2$

$x^2 = 64 + 36 = 100$

$x = \sqrt{100} = 10 \, \text{cm}$

Thus, if $x$ is the hypotenuse, its length is 10 cm.

2. $x$ as One of the Legs

If $x$ is one of the legs, we can use the Pythagorean theorem in reverse. Let’s assume $x$ is the unknown leg, with 8 cm as one leg and 6 cm as the other:

$x^2 + 6^2 = 8^2$

$x^2 + 36 = 64$

$x^2 = 64 - 36 = 28$

$x = \sqrt{28} \approx 5.29 \, \text{cm}$

Thus, if $x$ is one of the legs, its length is approximately 5.29 cm.

Possible Range for $x$

From the two cases, we conclude that the possible range for $x$ lies between 5.29 cm and 10 cm. Therefore, $x$ can be any value in this interval, depending on whether it's one of the legs or the hypotenuse.

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Would you like further details or clarifications? Here are a few related questions for you:

1. What is the general formula for the Pythagorean theorem?
2. How can we determine the perimeter of a right triangle?
3. What are the other properties of right triangles that might be useful here?
4. How do we apply the triangle inequality theorem to determine possible ranges for side lengths?
5. Can this concept be extended to non-right triangles?

Tip: Always remember that in any right triangle, the hypotenuse is always the longest side!