The radius of the circumcircle of a right angled triangle is 15 cm and the radio- of its inscribed circle is 6 cm. Find the sides of the triangle.

Options

A. A)30, 40, 41
B. B)18, 24, 30
C. C)30, 24, 25
D. D)24, 36, 20

Answer

To solve the problem, we need to find the sides of a right-angled triangle given the radius of the circumcircle (R) and the radius of the inscribed circle (r).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The radius of the circumcircle (R) = 15 cm.
- The radius of the inscribed circle (r) = 6 cm.

2. **Using the Relation for the Circumradius:**
- For a right-angled triangle, the circumradius (R) is given by the formula:
\[ R = \frac{H}{2} \]
where H is the hypotenuse of the triangle.
- Given R = 15 cm, we can find H:
\[ H = 2R = 2 \times 15 = 30 \text{ cm} \]

3. **Using the Relation for the Inradius:**
- The inradius (r) of a right-angled triangle can be expressed as:
\[ r = \frac{P + B - H}{2} \]
where P is the perpendicular side, B is the base, and H is the hypotenuse.
- Rearranging gives us:
\[ P + B = 2r + H \]
- Substituting the known values:
\[ P + B = 2 \times 6 + 30 = 12 + 30 = 42 \text{ cm} \]

4. **Setting Up the Equations:**
- We now have two equations:
1. \( P + B = 42 \) (1)
2. \( H = 30 \) (2)

5. **Using the Pythagorean Theorem:**
- For a right-angled triangle, we know:
\[ H^2 = P^2 + B^2 \]
- Substituting H = 30:
\[ 30^2 = P^2 + B^2 \]
\[ 900 = P^2 + B^2 \quad (3) \]

6. **Solving the System of Equations:**
- From equation (1), we can express B in terms of P:
\[ B = 42 - P \]
- Substitute B into equation (3):
\[ 900 = P^2 + (42 - P)^2 \]
- Expanding the equation:
\[ 900 = P^2 + (1764 - 84P + P^2) \]
\[ 900 = 2P^2 - 84P + 1764 \]
\[ 0 = 2P^2 - 84P + 864 \]
- Dividing the entire equation by 2:
\[ 0 = P^2 - 42P + 432 \]

7. **Using the Quadratic Formula:**
- We can solve for P using the quadratic formula:
\[ P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -42, c = 432 \):
\[ P = \frac{42 \pm \sqrt{(-42)^2 - 4 \cdot 1 \cdot 432}}{2 \cdot 1} \]
\[ P = \frac{42 \pm \sqrt{1764 - 1728}}{2} \]
\[ P = \frac{42 \pm \sqrt{36}}{2} \]
\[ P = \frac{42 \pm 6}{2} \]
- This gives us two possible values:
\[ P = \frac{48}{2} = 24 \quad \text{or} \quad P = \frac{36}{2} = 18 \]

8. **Finding B:**
- If \( P = 24 \):
\[ B = 42 - 24 = 18 \]
- If \( P = 18 \):
\[ B = 42 - 18 = 24 \]

9. **Conclusion:**
- The sides of the triangle are:
- Perpendicular (P) = 24 cm
- Base (B) = 18 cm
- Hypotenuse (H) = 30 cm

### Final Answer:
The sides of the triangle are 24 cm, 18 cm, and 30 cm.