If two sides of a triangles are 7 cm and 10 cm, then the largest possible integer value of the third side is _________.

Explanation

To find the largest possible integer value of the third side of a triangle when the other two sides are 7 cm and 10 cm, we 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.

### Step-by-Step Solution:

1. **Identify the given sides**: We have two sides of the triangle, which are 7 cm and 10 cm.

2. **Apply the triangle inequality theorem**: According to the theorem, for any triangle with sides \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

Here, let’s denote the third side as \( c \).

3. **Set up the inequalities**:
- From \( 7 + 10 > c \):
\[ 17 > c \quad \text{or} \quad c < 17 \]
- From \( 7 + c > 10 \):
\[ c > 3 \]
- From \( 10 + c > 7 \):
\[ c > -3 \quad \text{(this is always true since } c \text{ is a positive length)} \]

4. **Combine the inequalities**: The relevant inequalities we have are:
- \( c < 17 \)
- \( c > 3 \)

5. **Determine the largest integer value**: The largest integer value for \( c \) that satisfies \( c < 17 \) is 16.

### Final Answer:
The largest possible integer value of the third side is **16 cm**.

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