In the square $A B C D$ with side 1, point $F$ is the midpoint of side $B C$, and $E$ is the foot of the perpendicular dropped from vertex $A$ to $D F$.

Find the length of $B E$.

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Extend segment $D F$ to intersect line $A B$ at point $K$ (see figure). Right triangles $K B F$ and $D C F$ are equal (by leg and acute angle), so $K B = D C$. Therefore, $E B$ is the median of the right triangle $A E K$, drawn to the hypotenuse, hence $B E = 1 / 2 A K = 1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-39.jpg?height=597&width=355&top_left_y=525&top_left_x=843)

## Answer

1.