In a right triangle, the lengths of the sides are in arithmetic progression. If the lengths of the sides of the triangle are integers, which of the following could be the length of the shortest side ?

Options

A. 2125
B. 1700
C. 1275
D. 1150

Explanation

To solve the problem, we need to determine the possible lengths of the shortest side of a right triangle whose sides are in arithmetic progression and are integers.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In a right triangle, we denote the sides as \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse). Since the sides are in arithmetic progression, we can express them as:
\[ a = x - d, \quad b = x, \quad c = x + d \]
where \( x \) is the middle term and \( d \) is the common difference.

2. **Applying the Pythagorean Theorem**:
According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Substituting the expressions for \( a \), \( b \), and \( c \):
\[ (x - d)^2 + x^2 = (x + d)^2 \]

3. **Expanding the Equation**:
Expanding both sides:
\[ (x^2 - 2xd + d^2) + x^2 = (x^2 + 2xd + d^2) \]
Simplifying this gives:
\[ 2x^2 - 2xd + d^2 = x^2 + 2xd + d^2 \]

4. **Rearranging the Equation**:
Subtract \( x^2 + d^2 \) from both sides:
\[ 2x^2 - x^2 - 2xd - 2xd = 0 \]
This simplifies to:
\[ x^2 - 4xd = 0 \]
Factoring out \( x \):
\[ x(x - 4d) = 0 \]
This gives us two cases: \( x = 0 \) (not valid) or \( x = 4d \).

5. **Finding the Side Lengths**:
If \( x = 4d \), the sides become:
\[ a = 4d - d = 3d, \quad b = 4d, \quad c = 4d + d = 5d \]
Thus, the sides of the triangle are \( 3d, 4d, \) and \( 5d \).

6. **Identifying the Shortest Side**:
The shortest side is \( 3d \). Since \( d \) must be a positive integer, the shortest side can take values that are multiples of 3.

7. **Conclusion**:
Therefore, the possible lengths of the shortest side \( 3d \) can be any integer multiple of 3. If we are given options, we can check which of them is a multiple of 3.