ABC is a triangle, where BC = 2AB , `angle`B = `30^@` and `angle`A = `90^@`. The magnitude of the side AC is

Options

A. `(2BC)/3`
B. `(3BC)/4`
C. `(BC)/sqrt3`
D. `(sqrt3 BC)/(2)`

Answer

To find the magnitude of side AC in triangle ABC, where BC = 2AB, angle B = 30 degrees, and angle A = 90 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle Configuration**:
- Since angle A = 90 degrees, triangle ABC is a right triangle with angle B = 30 degrees. Therefore, angle C = 60 degrees (since the sum of angles in a triangle is 180 degrees).
- We can denote the sides as follows:
- AB = x (the side opposite angle C)
- AC = y (the side opposite angle B)
- BC = hypotenuse = 2AB = 2x

2. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem, in a right triangle:
\[ BC^2 = AB^2 + AC^2 \]
- Substituting the values we have:
\[ (2x)^2 = x^2 + y^2 \]
- This simplifies to:
\[ 4x^2 = x^2 + y^2 \]

3. **Rearrange the Equation**:
- Rearranging gives:
\[ 4x^2 - x^2 = y^2 \]
- Thus:
\[ 3x^2 = y^2 \]

4. **Solve for AC (y)**:
- Taking the square root of both sides:
\[ y = \sqrt{3}x \]

5. **Express AC in terms of BC**:
- Since BC = 2AB = 2x, we can express x in terms of BC:
\[ x = \frac{BC}{2} \]
- Substitute this back into the equation for AC:
\[ AC = \sqrt{3}x = \sqrt{3} \left(\frac{BC}{2}\right) = \frac{\sqrt{3}}{2} BC \]

### Final Answer:
The magnitude of side AC is:
\[ AC = \frac{\sqrt{3}}{2} BC \]