To solve for the height $h$ of the triangle:

Step-by-step:

1. Identify relevant triangle and relationships:

   • The given triangle has an angle of $45^\circ$, which forms a right triangle when considering the perpendicular height $h$.
   • The hypotenuse of this triangle is $34 \, \text{cm}$.
   • The angle adjacent to $h$ and the $45^\circ$ triangle is $105^\circ - 90^\circ = 15^\circ$.

2. Decompose the problem:

   • The side $h$ is opposite to the $45^\circ$ angle.
   • Use trigonometry to relate $h$ and the hypotenuse $34 \, \text{cm}$.

3. Use the sine function: For the triangle: $\sin(45^\circ) = \frac{h}{34}$ Therefore: $h = 34 \cdot \sin(45^\circ)$

4. Calculate $\sin(45^\circ)$: The exact value of $\sin(45^\circ)$ is: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ Substituting: $h = 34 \cdot \frac{\sqrt{2}}{2} = 17\sqrt{2} \, \text{cm}$

Final Answer:

$h = 17\sqrt{2} \, \text{cm}$

Would you like further explanation or more details on the steps?

Follow-Up Questions:

1. What is the relationship between trigonometric functions and right triangles?
2. How do you simplify expressions involving square roots?
3. Why is $\sin(45^\circ) = \frac{\sqrt{2}}{2}$?
4. Can you generalize this method for any angle in a triangle?
5. What other trigonometric ratios could be used to solve for $h$?

Tip:

Always label known sides and angles clearly when working with trigonometric problems—it simplifies identifying the appropriate formula.