In a `Delta ABC angle C = 60^(@) angle B = 90^(@) ` find the ratio of sides

Answer

To find the ratio of the sides of triangle ABC where angle C = 60° and angle B = 90°, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Angles of the Triangle:**
- Given: Angle B = 90° and Angle C = 60°.
- To find Angle A, use the property that the sum of angles in a triangle is 180°.
\[ \text{Angle A} = 180° - \text{Angle B} - \text{Angle C} = 180° - 90° - 60° = 30° \]

2. **Use the Sine Rule:**
- The Sine Rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]
where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles A, B, and C respectively.

3. **Substitute the Angles into the Sine Rule:**
- We know:
- \(A = 30°\)
- \(B = 90°\)
- \(C = 60°\)
- Thus, we can write:
\[ \frac{a}{\sin 30°} = \frac{b}{\sin 90°} = \frac{c}{\sin 60°} \]

4. **Calculate the Sine Values:**
- We know:
- \(\sin 30° = \frac{1}{2}\)
- \(\sin 90° = 1\)
- \(\sin 60° = \frac{\sqrt{3}}{2}\)

5. **Express Each Side in Terms of k:**
- From the Sine Rule:
\[ \frac{a}{\frac{1}{2}} = k \implies a = \frac{k}{2} \]
\[ \frac{b}{1} = k \implies b = k \]
\[ \frac{c}{\frac{\sqrt{3}}{2}} = k \implies c = \frac{\sqrt{3}k}{2} \]

6. **Form the Ratio of the Sides:**
- Now we have:
\[ a : b : c = \frac{k}{2} : k : \frac{\sqrt{3}k}{2} \]
- To simplify, we can factor out \(k\):
\[ = \frac{1}{2} : 1 : \frac{\sqrt{3}}{2} \]
- Multiply through by 2 to eliminate the fractions:
\[ = 1 : 2 : \sqrt{3} \]

### Final Answer:
The ratio of the sides of triangle ABC is:
\[ 1 : 2 : \sqrt{3} \]