The three sides of a triangle are 15, 25 and `x` units. Which one of the following is correct?

Options

A. ` 10 lt x lt 42`
B. `10 lt x lt 40`
C. ` 8 lt x lt 40`
D. ` 12 lt x lt 40`

Answer

To determine the possible values of \( x \) in the triangle with sides 15, 25, and \( x \), we will use the triangle inequality theorem. The triangle inequality 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 sides of the triangle**:
- Let the sides be \( a = 15 \), \( b = 25 \), and \( c = x \).

2. **Apply the triangle inequality theorem**:
- We need to satisfy the following inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

3. **Substituting the values into the inequalities**:
- For the first inequality:
\[ 15 + 25 > x \implies 40 > x \implies x < 40 \]
- For the second inequality:
\[ 15 + x > 25 \implies x > 25 - 15 \implies x > 10 \]
- For the third inequality:
\[ 25 + x > 15 \implies x > 15 - 25 \implies x > -10 \quad (\text{This inequality is always true since } x > 10) \]

4. **Combining the results**:
- From the first inequality, we have \( x < 40 \).
- From the second inequality, we have \( x > 10 \).
- Therefore, combining these results, we get:
\[ 10 < x < 40 \]

5. **Conclusion**:
- The value of \( x \) must be between 10 and 40. Thus, the correct option is that \( x \) is in between 10 to 40.