Question: h) 4 cm 75° 4 cm top-left: h) center: a quadrilateral/triangle-like line figure with a top vertex marked 75°, a left slanted side labeled 4 cm, and a right slanted side labeled 4 cm; a diagonal from bottom-left to right vertex crosses a vertical-ish segment from top to bottom with a right-angle marker at the intersection.
1. **Problem statement:**
We have a figure with two sides of length $4$ cm each and an angle of $75^\circ$ at the top vertex. We want to find the length $h$ (likely the height or a segment related to the figure).
2. **Understanding the figure:**
The figure appears to be an isosceles triangle with two equal sides of $4$ cm and an included angle of $75^\circ$ at the top.
3. **Formula used:**
To find the height $h$ from the top vertex to the base, we use the formula for the height in an isosceles triangle:
$$h = a \sin(\theta)$$
where $a$ is the length of the equal sides and $\theta$ is half the vertex angle.
4. **Calculate half the vertex angle:**
$$\theta = \frac{75^\circ}{2} = 37.5^\circ$$
5. **Calculate the height:**
$$h = 4 \sin(37.5^\circ)$$
6. **Evaluate $\sin(37.5^\circ)$:**
Using a calculator or sine table:
$$\sin(37.5^\circ) \approx 0.6088$$
7. **Calculate $h$ numerically:**
$$h = 4 \times 0.6088 = 2.4352$$
8. **Final answer:**
$$h \approx 2.44 \text{ cm}$$
This is the height from the top vertex perpendicular to the base in the given figure.