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 40`
B. `10 lt x lt 30 `
C. `20 lt x lt 40`
D. `9 lt xlt 40`

Answer

To determine the possible value of \( x \) in the triangle with sides 15, 25, and \( x \), we can use the triangle inequality theorem. This theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. We will apply this theorem to find the range of possible values for \( x \).

### Step-by-Step Solution:

1. **Identify the sides of the triangle**: We have three sides: 15, 25, and \( x \).

2. **Apply the triangle inequality theorem**:
- The first inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We can set up the following inequalities:
- \( 15 + 25 > x \)
- \( 15 + x > 25 \)
- \( 25 + x > 15 \)

3. **Solve 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:
\[ 25 + x > 15 \implies x > 15 - 25 \implies x > -10 \quad (\text{This is always true since } x > 10) \]

4. **Combine the results**: From the inequalities we derived, we have:
- \( x < 40 \)
- \( x > 10 \)

Therefore, the combined range for \( x \) is:
\[ 10 < x < 40 \]

5. **Conclusion**: The value of \( x \) must be greater than 10 and less than 40.