If 4cm and 7cm are the lengths of two sides of a triangle, find the possible length of the third side

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 the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, for a triangle with sides \(a\), \(b\), and \(c\), this can be expressed as:

\[
a + b > c
\]
\[
a + c > b
\]
\[
b + c > a
\]

Given that the lengths of two sides are 4 cm and 7 cm, let's denote the third side as \(x\).

Applying the Triangle Inequality Theorem:

1. \(4 + 7 > x\), so \(x < 11\)
2. \(4 + x > 7\), so \(x > 3\)
3. \(7 + x > 4\), so \(x > -3\) (but since side lengths must be positive, this inequality is redundant)

Thus, the possible length of the third side \(x\) must satisfy:
\[
3 < x < 11
\]

So, the third side must be **greater than 3 cm and less than 11 cm**.

Would you like more details or have any questions?

### Related Questions:
1. How does the Triangle Inequality Theorem apply to different types of triangles?
2. What are the consequences if the third side is exactly 3 cm or 11 cm?
3. How do you find the area of a triangle given all three sides?
4. Can a triangle exist with side lengths 4 cm, 7 cm, and 11 cm?
5. What is the relationship between the angles and sides in a triangle?

**Tip:** Always verify that a triangle with given side lengths satisfies the Triangle Inequality Theorem to ensure it's a valid triangle.

Discover how to determine the possible length of the third side of a triangle with sides 4 cm and 7 cm using the Triangle Inequality Theorem. This math problem explores geometric concepts and theorem applications. Suitable for Grades 7-9.

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