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

Options

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

Answer

To find the sides of the right-angled triangle given the circumradius (R) and inradius (r), we can use the following relationships:

1. The circumradius (R) of a right-angled triangle is given by the formula:
\[ R = \frac{c}{2} \]
where \(c\) is the length of the hypotenuse.

2. The inradius (r) of a right-angled triangle can be expressed in terms of the sides \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse) as:
\[ r = \frac{a + b - c}{2} \]

Given:
- \(R = 15\) cm
- \(r = 6\) cm

### Step 1: Find the hypotenuse \(c\)
Using the formula for the circumradius:
\[ R = \frac{c}{2} \implies c = 2R = 2 \times 15 = 30 \text{ cm} \]

### Step 2: Substitute \(c\) into the inradius formula
Now, substituting \(c\) into the inradius formula:
\[ r = \frac{a + b - c}{2} \]
Substituting the known values:
\[ 6 = \frac{a + b - 30}{2} \]

### Step 3: Solve for \(a + b\)
Multiply both sides by 2:
\[ 12 = a + b - 30 \]
Rearranging gives:
\[ a + b = 12 + 30 = 42 \]

### Step 4: Use the Pythagorean theorem
Since it is a right-angled triangle, we can use the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Substituting \(c = 30\):
\[ a^2 + b^2 = 30^2 = 900 \]

### Step 5: Solve the system of equations
Now we have a system of equations:
1. \(a + b = 42\)
2. \(a^2 + b^2 = 900\)

We can express \(b\) in terms of \(a\):
\[ b = 42 - a \]
Substituting into the second equation:
\[ a^2 + (42 - a)^2 = 900 \]
Expanding:
\[ a^2 + (1764 - 84a + a^2) = 900 \]
Combining like terms:
\[ 2a^2 - 84a + 1764 - 900 = 0 \]
\[ 2a^2 - 84a + 864 = 0 \]
Dividing the entire equation by 2:
\[ a^2 - 42a + 432 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ a = \frac{42 \pm \sqrt{(-42)^2 - 4 \cdot 1 \cdot 432}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{42 \pm \sqrt{1764 - 1728}}{2} \]
\[ = \frac{42 \pm \sqrt{36}}{2} \]
\[ = \frac{42 \pm 6}{2} \]
Calculating the two possible values for \(a\):
1. \(a = \frac{48}{2} = 24\)
2. \(a = \frac{36}{2} = 18\)

### Step 7: Find \(b\)
Using \(a + b = 42\):
1. If \(a = 24\), then \(b = 42 - 24 = 18\).
2. If \(a = 18\), then \(b = 42 - 18 = 24\).

Thus, the sides of the triangle are \(18\) cm and \(24\) cm, and the hypotenuse is \(30\) cm.

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