If sides of triangle are (8,b,14) then

Options

A. `8 lt b lt 14 `
B. `14 lt b lt 19`
C. `6 lt b lt 22`
D. ` 7 lt b lt 21 `

Answer

To determine the possible values of \( b \) in the triangle with sides \( 8, b, 14 \), we will use the triangle inequality theorem. The triangle inequality states that for any triangle with sides \( a, b, c \):

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

In our case, the sides are \( 8, b, \) and \( 14 \). We will apply the triangle inequality to find the range of values for \( b \).

### Step 1: Apply the triangle inequalities

1. **First inequality**: \( 8 + b > 14 \)
- Rearranging gives us:
\[ b > 14 - 8 \]
\[ b > 6 \]

2. **Second inequality**: \( 8 + 14 > b \)
- Rearranging gives us:
\[ b < 8 + 14 \]
\[ b < 22 \]

3. **Third inequality**: \( b + 14 > 8 \)
- Rearranging gives us:
\[ b > 8 - 14 \]
\[ b > -6 \]
(This condition is always satisfied since \( b > 6 \) is a stronger condition.)

### Step 2: Combine the inequalities

From the inequalities derived, we have:
- \( b > 6 \)
- \( b < 22 \)

Combining these gives us the range for \( b \):
\[ 6 < b < 22 \]

### Conclusion

Thus, the possible values for \( b \) are any real numbers between \( 6 \) and \( 22 \).

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