We are given a triangle $\Delta XYZ$ where:

• $x = 3.9 \, \text{cm}$ (the side opposite $\angle X$)
• $\angle Y = 68^\circ$
• $\angle Z = 86^\circ$

To find the length of side $z$ (the side opposite $\angle Z$), we can use the Law of Sines. The Law of Sines states:

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

In our case:

• $a = z$ (the side we are looking for, opposite $\angle Z$),
• $b = x = 3.9 \, \text{cm}$,
• $A = \angle Z = 86^\circ$,
• $B = \angle Y = 68^\circ$,
• $C = 180^\circ - 68^\circ - 86^\circ = 26^\circ$.

Now, applying the Law of Sines:

$\frac{z}{\sin 86^\circ} = \frac{3.9}{\sin 26^\circ}$

First, calculate the sines of the angles:

• $\sin 86^\circ \approx 0.9986$
• $\sin 26^\circ \approx 0.4384$

Now substitute these values:

$\frac{z}{0.9986} = \frac{3.9}{0.4384}$

Solve for $z$:

$z = \frac{3.9 \times 0.9986}{0.4384} \approx 8.88 \, \text{cm}$

Therefore, the length of side $z$ is approximately 8.9 cm (to the nearest tenth of a centimeter).

Would you like a breakdown of any steps? Here are five related questions:

1. How do you calculate the third angle in a triangle?
2. What is the Law of Cosines and when is it used?
3. How would you apply the Law of Sines if two sides and an angle were known?
4. Why is $\sin 86^\circ$ almost equal to 1?
5. How does the Law of Sines help in non-right triangles?

Tip: Always check that the sum of the angles in any triangle adds up to 180°.