If the measures of the angles of a triangle are in the ratio. 1:2:3 and if the length of the smallest side of the triangle is 10 cm, then the length of the longest side is

Options

A. 20 cm.
B. 25 cm.
C. 30 cm.
D. 35 cm.

Answer

To solve the problem, we need to determine the angles of the triangle based on the given ratio and then use the properties of triangles to find the length of the longest side.

### Step-by-Step Solution:

1. **Define the Angles**:
The angles of the triangle are in the ratio 1:2:3. Let's denote the angles as:
- Angle A = x
- Angle B = 2x
- Angle C = 3x

2. **Set Up the Equation**:
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ x + 2x + 3x = 180 \]

3. **Solve for x**:
Combine the terms:
\[ 6x = 180 \]
Now, divide both sides by 6:
\[ x = 30 \]

4. **Calculate the Angles**:
Now that we have x, we can find the measures of the angles:
- Angle A = x = 30 degrees
- Angle B = 2x = 60 degrees
- Angle C = 3x = 90 degrees

5. **Identify the Sides**:
Since the smallest side corresponds to the smallest angle, we know that the smallest side (10 cm) is opposite Angle A (30 degrees). The sides of the triangle will be in the ratio of the sine of the angles:
\[ \text{Side opposite A} : \text{Side opposite B} : \text{Side opposite C} = \sin(30^\circ) : \sin(60^\circ) : \sin(90^\circ) \]

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

7. **Set Up the Ratio**:
The sides can be expressed in terms of a common variable k:
- Side opposite A = \(10\) cm (smallest side)
- Side opposite B = \(10 \cdot \frac{\sin(60^\circ)}{\sin(30^\circ)} = 10 \cdot \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = 10\sqrt{3}\)
- Side opposite C = \(10 \cdot \frac{\sin(90^\circ)}{\sin(30^\circ)} = 10 \cdot \frac{1}{\frac{1}{2}} = 20\)

8. **Identify the Longest Side**:
The longest side is opposite the largest angle, which is Angle C (90 degrees). Therefore, the longest side is:
\[ \text{Longest side} = 20 \text{ cm} \]

### Final Answer:
The length of the longest side of the triangle is **20 cm**.