To find the length of side $z$ in triangle XYZ, we can use the Law of Sines.

The Law of Sines states: $\frac{\sin(\text{angle})}{\text{side opposite the angle}} = \frac{\sin(\text{angle})}{\text{side opposite the angle}} = \frac{\sin(\text{angle})}{\text{side opposite the angle}}$ In this case, we know:

• $\angle X = 40^\circ$
• $\angle Y = 70^\circ$
• $x = 15$ cm (side opposite $\angle X$)

Step 1: Find angle $Z$

Since the sum of the angles in a triangle is always 180°, we can find $\angle Z$ by subtracting the known angles from 180°: $\angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 40^\circ - 70^\circ = 70^\circ$

Step 2: Use the Law of Sines

We can now set up a proportion to find side $z$, which is opposite $\angle Z$: $\frac{x}{\sin(\angle X)} = \frac{z}{\sin(\angle Z)}$ Substituting the known values: $\frac{15}{\sin(40^\circ)} = \frac{z}{\sin(70^\circ)}$

Now, solve for $z$: $z = \frac{15 \cdot \sin(70^\circ)}{\sin(40^\circ)}$

Step 3: Calculate the result

First, calculate the sine values:

• $\sin(40^\circ) \approx 0.6428$
• $\sin(70^\circ) \approx 0.9397$

Substitute these into the equation: $z = \frac{15 \cdot 0.9397}{0.6428} \approx \frac{14.0955}{0.6428} \approx 21.91 \, \text{cm}$

So, the length of side $z$ is approximately 21.91 cm.

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Would you like a more detailed breakdown of the steps or have any further questions?

Here are 5 related questions for further exploration:

1. How do the angles in a triangle affect the length of the sides using the Law of Sines?
2. Can the Law of Cosines be used to solve this problem? How would that work?
3. What is the relationship between the angles and the sides in an isosceles triangle?
4. How can we determine whether a triangle is possible given side lengths and angles?
5. What would happen if the angles were different? How would the side lengths change?

Tip: When using trigonometric functions for triangle problems, always make sure your calculator is in the correct mode (degrees or radians) based on the angle units provided in the problem!