## Properties and characteristics of a parallelogram [Trigonometric ratios in a right triangle]

On the side $C D$ of parallelogram $A B C D$ with an obtuse angle at vertex $D$, an equilateral triangle $C D E$ is constructed such that points $A$ and $E$ lie on opposite sides of line $C D$. It is known that the distances from points $D$ and $E$ to line $B C$ are 3 and 8, respectively, and the distance from point $E$ to line $A B$ is 13. Find the area of parallelogram $A B C D$.
![](https://cdn.mathpix.com/cropped/2024_05_06_3af6d4d9a845a74ce61dg-08.jpg?height=1061&width=1884&top_left_y=1556&top_left_x=127)

Answer

$28 \sqrt{3}$.

Inside the parallelogram $K L M N$, a point $P$ is taken such that the triangle $K P N$ is equilateral. It is known that the distances from point $P$ to the lines $K L, L M$, and $M N$ are 10, 3, and 6, respectively. Find the perimeter of the parallelogram.

## Solution

![](https://cdn.mathpix.com/cropped/2024_05_06_3af6d4d9a845a74ce61dg-09.jpg?height=832&width=972&top_left_y=479&top_left_x=589)

## Answer

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