The sides of a right angled triangle arein `A.P.`, then they are in the ratio

Options

A. `2:3:1`
B. `2:3:5`
C. `3:4:5`
D. `3:1:2`

Answer

To find the ratio of the sides of a right-angled triangle that are in arithmetic progression (A.P.), we can follow these steps:

### Step-by-Step Solution:

1. **Define the sides of the triangle**:
Let the sides of the right-angled triangle be represented as \( a - d \), \( a \), and \( a + d \), where \( a \) is the middle term and \( d \) is the common difference.

2. **Use the Pythagorean theorem**:
Since it is a right-angled triangle, we can apply the Pythagorean theorem:
\[ (a + d)^2 = (a - d)^2 + a^2 \]

3. **Expand the equation**:
Expanding both sides gives:
\[ a^2 + 2ad + d^2 = (a^2 - 2ad + d^2) + a^2 \]

4. **Simplify the equation**:
Combine like terms:
\[ a^2 + 2ad + d^2 = 2a^2 - 2ad + d^2 \]
Subtract \( d^2 \) from both sides:
\[ a^2 + 2ad = 2a^2 - 2ad \]
Rearranging gives:
\[ a^2 - 4ad = 0 \]

5. **Factor the equation**:
Factoring out \( a \) gives:
\[ a(a - 4d) = 0 \]

6. **Solve for \( a \)**:
This results in two possible solutions: \( a = 0 \) or \( a = 4d \). Since \( a = 0 \) is not feasible for the sides of a triangle, we take \( a = 4d \).

7. **Determine the sides**:
Substituting \( a = 4d \) back into the expressions for the sides:
- First side: \( a - d = 4d - d = 3d \)
- Second side: \( a = 4d \)
- Third side: \( a + d = 4d + d = 5d \)

8. **Find the ratio of the sides**:
The sides of the triangle are \( 3d \), \( 4d \), and \( 5d \). Therefore, the ratio of the sides is:
\[ 3d : 4d : 5d = 3 : 4 : 5 \]

### Conclusion:
The sides of the right-angled triangle in arithmetic progression are in the ratio \( 3 : 4 : 5 \).

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