To calculate the missing side $x$ in the triangle, we can use the sine rule, which is given by:

$\frac{a}{\sin A} = \frac{b}{\sin B}$

Here, $a$, $b$ are sides of the triangle, and $A$, $B$ are their opposite angles. From the image:

• Side $6.5 \, \text{cm}$ is opposite the $71^\circ$ angle.
• Side $x$ is opposite the angle adjacent to $71^\circ$.

Step-by-Step Solution:

1. The third angle of the triangle: $\text{Angle opposite } x = 180^\circ - 90^\circ - 71^\circ = 19^\circ$

2. Using the sine rule: $\frac{6.5}{\sin 71^\circ} = \frac{x}{\sin 19^\circ}$

3. Rearrange to find $x$: $x = \frac{6.5 \cdot \sin 19^\circ}{\sin 71^\circ}$

4. Substitute values for $\sin 19^\circ$ and $\sin 71^\circ$:

   • $\sin 19^\circ \approx 0.32557$
   • $\sin 71^\circ \approx 0.94552$

   $x = \frac{6.5 \cdot 0.32557}{0.94552}$

5. Perform the calculation: $x \approx \frac{2.1162}{0.94552} \approx 2.24 \, \text{cm (to 2 decimal places)}.$

Final Answer:

The missing side $x$ is approximately 2.24 cm.

Would you like more details or have any questions?

Related Questions:

1. How is the sine rule different from the cosine rule in triangles?
2. Can this method be used for non-right triangles?
3. What are some practical examples where the sine rule is useful?
4. How can you verify the result of $x$ using trigonometric identities?
5. What happens if one of the angles of a triangle is obtuse?

Tip:

Always double-check angle values and calculations when working with trigonometry to avoid small but significant errors.