If the angles of a triangle ABC are in the ratio 1:2:3. then the corresponding sides are in the ratio ?

Options

A. `1:2:3`
B. `3:2:1`
C. `1:sqrt3:2`
D. `1:sqrt3:sqrt2`

Answer

To solve the problem of finding the ratio of the sides of triangle ABC given that the angles are in the ratio 1:2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles**:
Let the angles of triangle ABC be represented as:
- Angle A = x
- Angle B = 2x
- Angle C = 3x

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in a triangle is always 180 degrees. Therefore, we can set up the equation:
\[ x + 2x + 3x = 180 \]
This simplifies to:
\[ 6x = 180 \]

3. **Solve for x**:
Dividing both sides by 6 gives:
\[ x = 30 \]
Thus, the angles are:
- Angle A = 30 degrees
- Angle B = 60 degrees
- Angle C = 90 degrees

4. **Identify the Sides Using the Sine Rule**:
According to the sine rule, the ratio of the sides of a triangle is proportional to the sine of the opposite angles. Thus, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

5. **Substituting the Angles**:
We can express the sides in terms of the sine of the angles:
\[ \frac{a}{\sin 30^\circ} = \frac{b}{\sin 60^\circ} = \frac{c}{\sin 90^\circ} \]

6. **Calculate the Sine Values**:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\sin 90^\circ = 1\)

7. **Setting Up the Ratios**:
From the sine rule, we can set up the following ratios:
\[ \frac{a}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{3}}{2}} = \frac{c}{1} \]

8. **Expressing the Ratios**:
This gives us:
\[ a = k \cdot \frac{1}{2}, \quad b = k \cdot \frac{\sqrt{3}}{2}, \quad c = k \cdot 1 \]
where \(k\) is a constant.

9. **Finding the Ratio of the Sides**:
Therefore, the ratio of the sides \(a : b : c\) can be expressed as:
\[ a : b : c = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1 \]
Simplifying this gives:
\[ a : b : c = 1 : \sqrt{3} : 2 \]

### Final Answer:
The corresponding sides of triangle ABC are in the ratio \(1 : \sqrt{3} : 2\).