If the sides of a right angled triangle are in A.P., then `(R )/( r)=`

Options

A. `(5)/(2)`
B. `(7)/(3)`
C. `(9)/(4)`
D. `(8)/(3)`

Answer

To solve the problem, we need to find the ratio \( \frac{R}{r} \) for a right-angled triangle whose sides are in arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
Let the sides of the right-angled triangle be \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse). Since the sides are in A.P., we can express them as:
\[ a = A - d, \quad b = A, \quad c = A + d \]
Here, \( A \) is the middle term and \( d \) is the common difference.

2. **Use the Pythagorean theorem**:
For a right-angled triangle, the Pythagorean theorem states:
\[ c^2 = a^2 + b^2 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ (A + d)^2 = (A - d)^2 + A^2 \]

3. **Expand the equation**:
Expanding both sides:
\[ A^2 + 2Ad + d^2 = A^2 - 2Ad + d^2 + A^2 \]
Simplifying gives:
\[ A^2 + 2Ad + d^2 = 2A^2 - 2Ad + d^2 \]

4. **Rearranging the equation**:
Cancel \( d^2 \) from both sides:
\[ A^2 + 2Ad = 2A^2 - 2Ad \]
Rearranging leads to:
\[ 4Ad = A^2 \]
Thus, we find:
\[ A = 4d \]

5. **Calculate the semi-perimeter \( S \)**:
The semi-perimeter \( S \) is given by:
\[ S = \frac{a + b + c}{2} = \frac{(A - d) + A + (A + d)}{2} = \frac{3A}{2} = \frac{3(4d)}{2} = 6d \]

6. **Calculate the area \( \Delta \)**:
The area \( \Delta \) of the triangle can be calculated using:
\[ \Delta = \frac{1}{2} \times a \times b = \frac{1}{2} \times (A - d) \times A = \frac{1}{2} \times (4d - d) \times 4d = \frac{1}{2} \times 3d \times 4d = 6d^2 \]

7. **Calculate \( R \) and \( r \)**:
The circumradius \( R \) is given by:
\[ R = \frac{abc}{4\Delta} \]
Substituting the values:
\[ R = \frac{(A - d) \cdot A \cdot (A + d)}{4 \cdot 6d^2} = \frac{(3d)(4d)(5d)}{24d^2} = \frac{60d^3}{24d^2} = \frac{5d}{2} \]

The inradius \( r \) is given by:
\[ r = \frac{\Delta}{S} = \frac{6d^2}{6d} = d \]

8. **Find the ratio \( \frac{R}{r} \)**:
Now, we can find the ratio:
\[ \frac{R}{r} = \frac{\frac{5d}{2}}{d} = \frac{5}{2} \]

### Final Answer:
\[ \frac{R}{r} = \frac{5}{2} \]