[ Properties and characteristics of an isosceles triangle. ]

In trapezoid $ABCD$ ($AD$ is the larger base), diagonal $AC$ is perpendicular to side $CD$ and bisects angle $BAD$. It is known that $\angle CDA = 60^\circ$, and the perimeter of the trapezoid is 2. Find $AD$.
The lateral sides of the trapezoid and the smaller base are equal to each other.

## Solution

Since $\angle C A D=90^{\circ}-\angle C D A=30^{\circ}, \angle A C B=\angle C A D=\angle B A C=30^{\circ}$, then $A B=B C, \angle B A D=60^{\circ}, C D=A B=$ $B C, A D=2 C D$. Therefore, $5 C D=2, C D=2 / 5, A D=4 / 5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-27.jpg?height=426&width=861&top_left_y=256&top_left_x=610)

## Answer

$4 / 5$

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