3.68. The lateral sides of the trapezoid are equal to $p$ and $q(p<q)$, the larger base is equal to $a$. The angles at the larger base are in the ratio $2: 1$. Find the smaller base.
3.68. I n d i c a t i o n. D e n o t i n g t h e a n g l e s a t t h e l a r g e r b a s e o f t h e t r a p e z o i d a s $\alpha$ a n d $2 \alpha$, d r o p t h e h e i g h t s o f t h e t r a p e z o i d o n t o t h i s b a s e a n d u s e t h e i r e q u a l i t y.

A n s w e r: $\frac{p^{2}+a p-q^{2}}{p}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-543.jpg?height=277&width=525&top_left_y=1306&top_left_x=45)

F i g. 3.70