If `Delta`ABC is right angled at B, AB = 30 and `angleACB = 60^@`, then what is the value of AC?

Options

A. 20
B. `20sqrt()3`
C. `40`
D. 60

Answer

To solve the problem, we need to find the length of side AC in triangle ABC, which is a right triangle with the right angle at B. We are given that AB = 30 cm and angle ACB = 60 degrees.

### Step-by-Step Solution:

1. **Identify the triangle and its angles**:
- Triangle ABC is a right triangle with angle B = 90 degrees.
- We know that angle ACB = 60 degrees.
- Therefore, angle CAB can be calculated as:
\[ \text{Angle CAB} = 90^\circ - 60^\circ = 30^\circ \]

2. **Label the sides**:
- Let AB = c = 30 cm (the side opposite angle ACB).
- Let BC = a (the side opposite angle CAB).
- Let AC = b (the hypotenuse opposite angle B).

3. **Use the properties of a right triangle**:
- In a right triangle, the sides opposite the angles have specific ratios. For angle ACB (60 degrees), the ratio of the sides is:
\[ \text{Opposite (AB)} : \text{Adjacent (BC)} : \text{Hypotenuse (AC)} = 1 : \sqrt{3} : 2 \]
- Here, the side opposite angle ACB (60 degrees) is AB, which is 30 cm.

4. **Set up the ratio**:
- Since AB is opposite the 60-degree angle, we can set up the ratio:
\[ \frac{AB}{AC} = \frac{1}{2} \]
- Plugging in the known value:
\[ \frac{30}{AC} = \frac{1}{2} \]

5. **Solve for AC**:
- Cross-multiplying gives:
\[ 30 \cdot 2 = AC \cdot 1 \]
\[ AC = 60 \text{ cm} \]

6. **Use the sine rule to confirm**:
- Alternatively, we can use the sine function:
\[ \sin(60^\circ) = \frac{AB}{AC} \]
- Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ \frac{\sqrt{3}}{2} = \frac{30}{AC} \]
- Cross-multiplying gives:
\[ AC \cdot \sqrt{3} = 60 \]
\[ AC = \frac{60}{\sqrt{3}} = 20\sqrt{3} \text{ cm} \]

### Final Answer:
The value of AC is \(20\sqrt{3}\) cm.