Given the sides of a triangle as 3.4cm and 5.2 cm, what can be the length of the third side (x) in cm?

Options

A. `1.8 ltxlt8.6`
B. `gt8.6`
C. `lt1.8`
D. `3.4lt xlt5.2`

Answer

To find the possible length of the third side \( x \) of a triangle given the lengths of the other two sides (3.4 cm and 5.2 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, and the difference of the lengths of any two sides must be less than the length of the third side.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
Let the sides be \( a = 3.4 \) cm and \( b = 5.2 \) cm. We need to find the possible values for the third side \( x \).

2. **Apply the triangle inequality theorem**:
According to the theorem, we have two conditions to consider:
- The sum of the two sides must be greater than the third side:
\[ a + b > x \]
- The difference of the two sides must be less than the third side:
\[ |a - b| < x \]

3. **Calculate the sum of the two sides**:
\[ 3.4 + 5.2 = 8.6 \]
So, we have:
\[ x < 8.6 \]

4. **Calculate the difference of the two sides**:
\[ 5.2 - 3.4 = 1.8 \]
Thus, we have:
\[ x > 1.8 \]

5. **Combine the inequalities**:
From the calculations, we have:
\[ 1.8 < x < 8.6 \]

### Conclusion:
The length of the third side \( x \) must be greater than 1.8 cm and less than 8.6 cm.