The side of a triangle are distinct integers in an arithmetic progression. If the smallest side is 10, the number of such triangles is-

Options

A. 8
B. 9
C. 10
D. Infinitely many

Explanation

To solve the problem, we need to determine the number of distinct triangles that can be formed with sides in an arithmetic progression (AP) where the smallest side is 10.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
Let the sides of the triangle be \( a \), \( b \), and \( c \) such that \( a < b < c \). Given that the smallest side \( a = 10 \), we can express the other sides in terms of a common difference \( d \):
- \( a = 10 \)
- \( b = 10 + d \)
- \( c = 10 + 2d \)

2. **Apply the triangle inequality**:
For a set of lengths to form a triangle, they must satisfy the triangle inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

We will check these inequalities one by one.

3. **Check the first inequality \( a + b > c \)**:
\[ 10 + (10 + d) > (10 + 2d) \]
Simplifying this gives:
\[ 20 + d > 10 + 2d \implies 20 - 10 > 2d - d \implies 10 > d \]
Thus, \( d < 10 \).

4. **Check the second inequality \( a + c > b \)**:
\[ 10 + (10 + 2d) > (10 + d) \]
Simplifying this gives:
\[ 20 + 2d > 10 + d \implies 20 - 10 > d \implies 10 > d \]
This condition is already satisfied since \( d < 10 \).

5. **Check the third inequality \( b + c > a \)**:
\[ (10 + d) + (10 + 2d) > 10 \]
Simplifying this gives:
\[ 20 + 3d > 10 \implies 3d > -10 \]
This condition is always true for positive \( d \).

6. **Determine the possible integer values for \( d \)**:
Since \( d \) must be a positive integer and \( d < 10 \), the possible values for \( d \) are:
\[ d = 1, 2, 3, 4, 5, 6, 7, 8, 9 \]
This gives us a total of 9 distinct values for \( d \).

### Conclusion:
The number of distinct triangles that can be formed with sides in an arithmetic progression, where the smallest side is 10, is **9**.