In a right-angled triangle $ riangle ABC$ right angled at $B$, given $AB = 12 ext{ cm}$ and $\angle CAB = 60^\circ$, determine the length of $BC$.

Question

In a right-angled triangle $	riangle ABC$ right angled at $B$, given $AB = 12 	ext{ cm}$ and $\angle CAB = 60^\circ$, determine the length of $BC$.

Accepted Answer

**Solution:**

1. Given $	riangle ABC$ is right angled at $B$, with $AB = 12 	ext{ cm}$ and $\angle CAB = 60^\circ$.

2. We need to find the length of $BC$.

3. Since $\angle CAB = 60^\circ$, the other acute angle at $C$ is $30^\circ$ because the triangle is right angled at $B$ and angles sum to $180^\circ$.

4. Using the angle $30^\circ$ at $C$, consider the tangent ratio:
   $$	an 30^\circ = \frac{	ext{opposite side}}{	ext{adjacent side}} = \frac{AB}{BC}$$

5. We know $	an 30^\circ = \frac{1}{\sqrt{3}}$ and $AB = 12$ cm.

6. Substitute values:
   $$\frac{1}{\sqrt{3}} = \frac{12}{BC} \implies BC = 12 \sqrt{3} 	ext{ cm}$$

**Answer:** $12\sqrt{3} 	ext{ cm}$

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