If the angles of a triangle are in the ratio 1:2:3,the corresponding sides are in the ratio

Options

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

Answer

To solve the problem of finding the ratio of the sides of a triangle when 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 the triangle be represented as:
- Angle A = θ
- Angle B = 2θ
- Angle C = 3θ

2. **Use the Angle Sum Property**:
The sum of the angles in a triangle is always 180 degrees (or π radians). Therefore, we can write the equation:
\[ A + B + C = θ + 2θ + 3θ = 6θ = π \]

3. **Solve for θ**:
To find the value of θ, we rearrange the equation:
\[ 6θ = π \implies θ = \frac{π}{6} \]

4. **Calculate Each Angle**:
Now we can find the individual angles:
- Angle A = θ = \(\frac{π}{6}\)
- Angle B = 2θ = \(2 \times \frac{π}{6} = \frac{π}{3}\)
- Angle C = 3θ = \(3 \times \frac{π}{6} = \frac{π}{2}\)

5. **Use the Sine Rule**:
According to the sine rule, the ratio of the sides opposite to the angles is equal to the ratio of the sines of those angles:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Thus, we can express the ratios of the sides as:
\[ a : b : c = \sin A : \sin B : \sin C \]

6. **Calculate the Sines of the Angles**:
Now we calculate the sine of each angle:
- \(\sin A = \sin\left(\frac{π}{6}\right) = \frac{1}{2}\)
- \(\sin B = \sin\left(\frac{π}{3}\right) = \frac{\sqrt{3}}{2}\)
- \(\sin C = \sin\left(\frac{π}{2}\right) = 1\)

7. **Form the Ratios**:
Now we can write the ratios of the sides:
\[ a : b : c = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1 \]
To eliminate the fractions, we can multiply each term by 2:
\[ a : b : c = 1 : \sqrt{3} : 2 \]

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

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