In a trianlge ABC if `A=90^(@), b =3 and c=4` then R : r is

Options

A. `5:3`
B. `7:3`
C. `3:2`
D. `5:2`

Answer

To solve the problem, we need to find the ratio of the circumradius (R) to the inradius (r) of triangle ABC, where angle A is 90 degrees, side b = 3, and side c = 4.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
Given that angle A = 90°, we can identify the sides of the triangle:
- Side opposite angle A (which is side BC) is the hypotenuse.
- Let AB = c = 4 (the side opposite angle B).
- Let AC = b = 3 (the side opposite angle C).
- Therefore, BC = hypotenuse = √(AB² + AC²) = √(4² + 3²) = √(16 + 9) = √25 = 5.

2. **Calculate the circumradius (R)**:
The formula for the circumradius (R) of a right triangle is given by:
\[ R = \frac{c}{2} \]
where c is the length of the hypotenuse.
Substituting the value of c:
\[ R = \frac{5}{2} \]

3. **Calculate the inradius (r)**:
The formula for the inradius (r) of a right triangle is given by:
\[ r = \frac{a + b - c}{2} \]
where a and b are the lengths of the two legs, and c is the hypotenuse.
Here, a = 3, b = 4, and c = 5.
Substituting the values:
\[ r = \frac{3 + 4 - 5}{2} = \frac{2}{2} = 1 \]

4. **Find the ratio R : r**:
Now we can find the ratio of the circumradius to the inradius:
\[ \frac{R}{r} = \frac{\frac{5}{2}}{1} = \frac{5}{2} \]

5. **Final Ratio**:
Therefore, the ratio R : r is 5 : 2.

### Conclusion:
The answer is that the ratio of the circumradius to the inradius (R : r) is 5 : 2.