Choose the CORRECT option if the two sides of a triangle is of length 4 cm and 10 cm and the length of its third side is a cm.

Options

A. `6 lt a lt 14`
B. `6`
C. `14 gt a gt 5`
D. `5 lt a lt 12`

Answer

To solve the problem, we need to apply the triangle inequality theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

In this case, we are given two sides of the triangle: \( 4 \) cm and \( 10 \) cm, and we need to find the possible range for the third side \( a \).

### Step 1: Apply the triangle inequality theorem

Let's denote the sides as follows:
- Side 1: \( 4 \) cm
- Side 2: \( 10 \) cm
- Side 3: \( a \) cm

Using the triangle inequality, we can derive the following inequalities:

1. \( 4 + 10 > a \)
2. \( 4 + a > 10 \)
3. \( 10 + a > 4 \)

### Step 2: Solve each inequality

1. From \( 4 + 10 > a \):
\[ 14 > a \quad \text{or} \quad a < 14 \]

2. From \( 4 + a > 10 \):
\[ a > 10 - 4 \quad \Rightarrow \quad a > 6 \]

3. From \( 10 + a > 4 \):
\[ a > 4 - 10 \quad \Rightarrow \quad a > -6 \quad \text{(This inequality is always true since \( a \) must be positive)} \]

### Step 3: Combine the inequalities

From the inequalities derived, we have:
- \( a > 6 \)
- \( a < 14 \)

Thus, the range for the length of the third side \( a \) is:
\[ 6 < a < 14 \]

### Conclusion

The correct option for the length of the third side \( a \) must be between \( 6 \) cm and \( 14 \) cm.