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The problem asks to find the range of possible lengths for the third side of a triangle when the two given sides are 6 cm and 10 cm, respectively. This question relates to the **triangle inequality theorem**, which states that for any triangle with sides \( a \), \( b \), and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, let \( x \) represent the length of the third side. The sides given are 6 cm and 10 cm.

### Applying the triangle inequality theorem:

1. \( 6 + 10 > x \Rightarrow x < 16 \)
2. \( 6 + x > 10 \Rightarrow x > 4 \)
3. \( 10 + x > 6 \) (this condition is always true, so no new restriction here)

Thus, the length of the third side must satisfy:
\[ 4 < x < 16 \]

The possible range of lengths for the third side is \( (4, 16) \) cm.

Would you like more details on this? Let me know if you have any questions. 

### Related questions:
1. What is the triangle inequality theorem and why is it important?
2. How do you determine whether three given lengths can form a triangle?
3. How would the range of the third side change if the given sides were different?
4. What happens if one of the sides is longer than the sum of the other two?
5. Can a triangle have sides that violate the triangle inequality?

**Tip:** Always check the triangle inequality to ensure that a valid triangle can be formed with given side lengths.

Two sides of a given triangle are 6 cm and 10 cm. Find the range of possible lengths of the third side of the triangle.

Discover how to apply the triangle inequality theorem to determine the possible range of lengths for the third side of a triangle with sides of 6 cm and 10 cm. This problem covers key geometric concepts and is suitable for students in Grades 7-9.

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