In a rectangle ABCD, AB= 15 cm `angle BAC = 60^@` then BC is equal to

Options

A. `15 sqrt(3)`
B. `(15sqrt(3))/(2)`
C. `(15)/(sqrt(3))`
D. `(15)/(2 sqrt(3))`

Answer

To solve the problem, we need to find the length of side BC in rectangle ABCD, given that AB = 15 cm and angle BAC = 60°.

### Step-by-step Solution:

1. **Identify the triangle**: In rectangle ABCD, we can focus on triangle ABC. Here, AB is one side, and we need to find the length of side BC.

2. **Use the tangent function**: We know that in triangle ABC, angle BAC is given as 60°. We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle. The formula for tangent is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case, angle BAC = 60°, the opposite side is BC, and the adjacent side is AB.

3. **Substitute the known values**: We can substitute the values we have into the tangent formula:
\[ \tan(60°) = \frac{BC}{AB} \]
Given that AB = 15 cm, we have:
\[ \tan(60°) = \frac{BC}{15} \]

4. **Calculate tan(60°)**: The value of \(\tan(60°)\) is \(\sqrt{3}\). Therefore, we can rewrite the equation:
\[ \sqrt{3} = \frac{BC}{15} \]

5. **Solve for BC**: To find BC, we can multiply both sides by 15:
\[ BC = 15 \cdot \sqrt{3} \]

6. **Final answer**: Thus, the length of BC is:
\[ BC = 15\sqrt{3} \text{ cm} \]

### Summary:
The length of side BC in rectangle ABCD is \(15\sqrt{3}\) cm.