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

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the sides of the triangle
Let the sides of the right-angled triangle be in arithmetic progression. We can denote the sides as:
- Smallest side: \( a - d \)
- Middle side: \( a \)
- Largest side (hypotenuse): \( a + d \)

### Step 2: Use the area of the triangle
The area of a right-angled triangle is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, we can take the base as the smallest side \( (a - d) \) and the height as the middle side \( a \). Given that the area is 24, we can set up the equation:
\[ \frac{1}{2} \times (a - d) \times a = 24 \]
Multiplying both sides by 2 gives:
\[ (a - d) \times a = 48 \]
This simplifies to:
\[ a^2 - ad = 48 \quad \text{(Equation 1)} \]

### Step 3: Apply Pythagoras' theorem
According to Pythagoras' theorem, we have:
\[ \text{(hypotenuse)}^2 = \text{(base)}^2 + \text{(height)}^2 \]
Substituting the sides, we get:
\[ (a + d)^2 = (a - d)^2 + a^2 \]
Expanding both sides:
\[ a^2 + 2ad + d^2 = a^2 - 2ad + d^2 + a^2 \]
This simplifies to:
\[ a^2 + 2ad + d^2 = 2a^2 - 2ad + d^2 \]
Cancelling \( d^2 \) from both sides:
\[ a^2 + 2ad = 2a^2 - 2ad \]
Rearranging gives:
\[ 4ad = a^2 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
From Equation 2, we can express \( a \) in terms of \( d \):
\[ a = 4d \]
Now substitute \( a = 4d \) into Equation 1:
\[ (4d)^2 - (4d)d = 48 \]
This simplifies to:
\[ 16d^2 - 4d^2 = 48 \]
\[ 12d^2 = 48 \]
Dividing both sides by 12:
\[ d^2 = 4 \]
Taking the square root:
\[ d = 2 \quad (\text{since } d \text{ must be positive}) \]

### Step 5: Find the value of \( a \)
Now substitute \( d = 2 \) back into the equation for \( a \):
\[ a = 4d = 4 \times 2 = 8 \]

### Step 6: Calculate the smallest side
The smallest side is given by:
\[ \text{Smallest side} = a - d = 8 - 2 = 6 \]

### Final Answer
The length of the smallest side is \( \boxed{6} \).