1.30. The lengths of the parallel sides of the trapezoid are 25 and 4 cm, and the lengths of the non-parallel sides are 20 and 13 cm. Find the height of the trapezoid.
1.30. Given that $B C=$ $=4 \text{ cm}, A D=25 \text{ cm}, A B=20$ cm, $C D=13$ cm (Fig. 1.26). Draw $B E \perp A D$ and $C F \perp A D$. Let $B E=C F=$ $=h, A E=x, F D=y$. Then from $\triangle A B E$ and $\triangle C F D$ we find $h=\sqrt{20^{2}-x^{2}}=\sqrt{13^{2}-y^{2}}$. Considering that $y=25-4-$ $-x=21-x$, we have

$$
20^{2}-x^{2}=13^{2}-(21-x)^{2} \text {, or } 42 x=672 \text {, }
$$

from which $x=16$ (cm). Thus, $h=\sqrt{20^{2}-16^{2}}=12$ (cm).

Answer: $12 \text{ cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-427.jpg?height=511&width=512&top_left_y=1152&top_left_x=46)

Fig. 1.27