To determine the possible length of the third side of a triangle when two sides are given, you can use the triangle inequality theorem. This theorem states that for any triangle, the length of any side must be greater than the difference and less than the sum of the lengths of the other two sides.

Given:

• Side 1 = 4 cm
• Side 2 = 7 cm

Let the third side be $x$ cm.

According to the triangle inequality theorem, the following inequalities must be satisfied:

1. $x + 4 > 7$
2. $x + 7 > 4$
3. $4 + 7 > x$

Let's solve these inequalities:

1. $x + 4 > 7$ $x > 7 - 4$ $x > 3$

2. $x + 7 > 4$ $x > 4 - 7$ x > -3 \quad (\text{This is always true since x is positive.})

3. $4 + 7 > x$ $11 > x$ $x < 11$

Therefore, the possible length of the third side $x$ must satisfy: $3 < x < 11$

Conclusion: The length of the third side must be greater than 3 cm and less than 11 cm.

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. If the two sides are 5 cm and 8 cm, what is the range for the third side?
2. Can a triangle have sides of lengths 2 cm, 7 cm, and 9 cm?
3. What happens to the range of the third side if one of the given sides increases?
4. If the third side is exactly 7 cm, what type of triangle would this form?
5. How does the triangle inequality theorem apply in the context of an equilateral triangle?

Tip: The triangle inequality theorem is fundamental in determining whether a given set of side lengths can form a valid triangle. Always check these inequalities when dealing with triangles.