The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Explanation

To solve the problem, we need to find the smallest side of a right-angled triangle whose sides are in arithmetic progression and whose area is 24. Let's denote the sides of the triangle as \( a - d \), \( a \), and \( a + d \), where \( a \) is the middle term and \( d \) is the common difference.

### Step 1: Set up the equation for the area of the triangle
The area \( A \) of a right-angled triangle can be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, we can take the two shorter sides as the base and height. Thus, we have:
\[ A = \frac{1}{2} \times (a - d) \times a = 24 \]
This simplifies to:
\[ (a - d) \times a = 48 \]

### Step 2: Use the Pythagorean theorem
Since the triangle is a right triangle, we can apply the Pythagorean theorem:
\[ (a - d)^2 + a^2 = (a + d)^2 \]
Expanding this gives:
\[ (a^2 - 2ad + d^2) + a^2 = (a^2 + 2ad + d^2) \]
This simplifies to:
\[ 2a^2 - 2ad = a^2 + 2ad \]
Rearranging terms, we get:
\[ a^2 - 4ad = 0 \]
Factoring out \( a \), we find:
\[ a(a - 4d) = 0 \]
Since \( a \) cannot be zero (as it represents a side length), we have:
\[ a = 4d \]

### Step 3: Substitute \( a \) back into the area equation
Now we substitute \( a = 4d \) back into the area equation:
\[ (4d - d) \times 4d = 48 \]
This simplifies to:
\[ 3d \times 4d = 48 \]
\[ 12d^2 = 48 \]
Dividing both sides by 12 gives:
\[ d^2 = 4 \]
Taking the square root, we find:
\[ d = 2 \]

### Step 4: Find the lengths of the sides
Now we can find the lengths of the sides:
- The smallest side: \( a - d = 4d - d = 3d = 3 \times 2 = 6 \)
- The middle side: \( a = 4d = 4 \times 2 = 8 \)
- The largest side: \( a + d = 4d + d = 5d = 5 \times 2 = 10 \)

### Conclusion
Thus, the lengths of the sides of the triangle are 6, 8, and 10, and the smallest side is:
\[ \text{Smallest side} = 6 \]