If the angles of a triangle are in the ratio `1:2:3` and its circumradius is 10 cm, find the sides of the triangle (in cm):

Options

A. 8, 16, 24
B. 18, 12, 6
C. `10, 20, 20sqrt3`
D. `10, 20, 10sqrt3`

Answer

To find the sides of a triangle given that its angles are in the ratio of 1:2:3 and its circumradius is 10 cm, we can follow these steps:

### Step 1: Determine the Angles of the Triangle
Let the angles of the triangle be represented as:
- Angle A = k
- Angle B = 2k
- Angle C = 3k

Since the sum of the angles in a triangle is 180 degrees, we can write the equation:
\[ k + 2k + 3k = 180 \]

### Step 2: Solve for k
Combine the terms:
\[ 6k = 180 \]
Now, divide both sides by 6:
\[ k = \frac{180}{6} = 30 \text{ degrees} \]

### Step 3: Find Each Angle
Now we can find the individual angles:
- Angle A = k = 30 degrees
- Angle B = 2k = 2 * 30 = 60 degrees
- Angle C = 3k = 3 * 30 = 90 degrees

### Step 4: Identify the Type of Triangle
Since one of the angles is 90 degrees, this is a right-angled triangle.

### Step 5: Use the Circumradius to Find the Hypotenuse
For a right-angled triangle, the circumradius (R) is given by:
\[ R = \frac{c}{2} \]
where c is the hypotenuse.

Given that the circumradius is 10 cm:
\[ 10 = \frac{c}{2} \]
Thus, the hypotenuse (c) is:
\[ c = 10 \times 2 = 20 \text{ cm} \]

### Step 6: Use Trigonometric Ratios to Find the Other Sides
Now we can find the other two sides using trigonometric ratios.

#### Finding Side AB (opposite to angle B, which is 60 degrees):
Using the sine function:
\[ \sin(60) = \frac{AB}{c} \]
Substituting the values:
\[ \sin(60) = \frac{AB}{20} \]
We know that \( \sin(60) = \frac{\sqrt{3}}{2} \):
\[ \frac{\sqrt{3}}{2} = \frac{AB}{20} \]
Now, solve for AB:
\[ AB = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \text{ cm} \]

#### Finding Side BC (adjacent to angle B, which is 60 degrees):
Using the cosine function:
\[ \cos(60) = \frac{BC}{c} \]
Substituting the values:
\[ \cos(60) = \frac{BC}{20} \]
We know that \( \cos(60) = \frac{1}{2} \):
\[ \frac{1}{2} = \frac{BC}{20} \]
Now, solve for BC:
\[ BC = 20 \times \frac{1}{2} = 10 \text{ cm} \]

### Final Result
The sides of the triangle are:
- AB = \( 10\sqrt{3} \) cm
- BC = 10 cm
- AC (hypotenuse) = 20 cm

### Summary of Sides
- Side AB = \( 10\sqrt{3} \) cm
- Side BC = 10 cm
- Side AC = 20 cm