10.057. The diagonal of a rectangular trapezoid and its lateral side are equal. Find the length of the midline if the height of the trapezoid is 2 cm and the lateral side is $4 \mathrm{cm}$.
Solution.

Given $A D=2$ cm; $D B=B C=4$ cm (Fig. 10.56). In $\triangle D A B$, $\angle D A B=90^{\circ}$, $A B=\sqrt{16-4}=\sqrt{12}$, $B E=\frac{B K}{2}=\frac{A D}{2}=\frac{2}{2}=1$, $B M=\frac{B C}{2}=2$. In $\triangle B E M$, $\angle B E M=90^{\circ}$, $E M=\sqrt{4-1}=\sqrt{3}$, and $N M=N E+E M=A B+E M=$ $=\sqrt{12}+\sqrt{3}=3 \sqrt{3}(\mathrm{~cm})$.

Answer: $3 \sqrt{3} \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-583.jpg?height=450&width=408&top_left_y=95&top_left_x=250)

Fig. 10.57

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-583.jpg?height=350&width=322&top_left_y=161&top_left_x=767)

Fig. 10.58