The length of two sides of a triangle are 7 cm and 9 cm. The length of the third side may lie between

Options

A. 1 cm and 10 cm
B. 2 cm and 8 cm
C. 3 cm and 16 cm
D. 1 cm and 16 cm

Answer

To determine the possible lengths of the third side of a triangle when the lengths of the other two sides are given as 7 cm and 9 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 lengths of the two known sides be \( a = 7 \) cm and \( b = 9 \) cm. We need to find the range for the length of the third side, which we will denote as \( x \).

2. **Apply the triangle inequality (sum condition)**:
According to the triangle inequality, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we can write:
\[ a + b > x \]
Substituting the known values:
\[ 7 + 9 > x \\ 16 > x \\ x < 16 \]

3. **Apply the triangle inequality (difference condition)**:
The difference between the lengths of any two sides must be less than the length of the third side. Thus, we have:
\[ |a - b| < x \]
Calculating the difference:
\[ |7 - 9| < x \\ 2 < x \]

4. **Combine the inequalities**:
From the two inequalities we derived:
- \( x < 16 \)
- \( x > 2 \)

We can combine these to find the range of \( x \):
\[ 2 < x < 16 \]

5. **Conclusion**:
The length of the third side \( x \) must be greater than 2 cm and less than 16 cm. Therefore, the possible range for the length of the third side is:
\[ (2 \text{ cm}, 16 \text{ cm}) \]

### Final Answer:
The length of the third side may lie between 2 cm and 16 cm.

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