To find the height $h$ of the triangle, we can use trigonometric relationships within the given triangle.

Step-by-step solution:

1. Given information:

   • The side of the triangle has a length of 34 cm.
   • The angle opposite the height $h$ is $45^\circ$.
   • The angle between the given side and the base is $105^\circ$.

2. Using the sine rule (Law of Sines): In this triangle, the height $h$ forms a right triangle with the angle $45^\circ$ at the bottom right. We can use the sine of the angle $45^\circ$ to find the height $h$.

   From the right triangle: $\sin(45^\circ) = \frac{h}{34}$

3. Solving for $h$: Using the known value for $\sin(45^\circ) = \frac{\sqrt{2}}{2}$: $\frac{\sqrt{2}}{2} = \frac{h}{34}$

   Multiply both sides by 34: $h = 34 \times \frac{\sqrt{2}}{2}$ $h = 17\sqrt{2}$

Thus, the height $h$ is $17\sqrt{2} \, \text{cm}$ in its simplest form.

Would you like more details or have any questions?

Related Questions:

1. How is the sine rule different from the cosine rule?
2. Can we find the other sides of this triangle using trigonometry?
3. What is the significance of the angle $105^\circ$ in this triangle?
4. How would the height change if the angle at the bottom right was different?
5. Could we solve this using the Law of Cosines instead?

Tip:

Always check if there is a simpler right-triangle approach for problems like this before using more complex laws like sine or cosine rules!