A4. In rectangle $A B C D$, $S$ is the intersection of the diagonals. E lies on $A C$ between $A$ and $S$. $D E$ is perpendicular to AS.

$D S=5, S E=3$ and $E D=4$.

Determine the length of $A B$.

![](https://cdn.mathpix.com/cropped/2024_04_17_9dc9851d634283e8a951g-2.jpg?height=308&width=531&top_left_y=206&top_left_x=1342)
A4. $D S=A S$ so $A E=2 . A D^{2}=4^{2}+2^{2}=20$.

Consider triangle $A B D$. $B D=10$, so $B D^{2}=100$.

Using Pythagoras, we find directly:

$A B=\sqrt{B D^{2}-A D^{2}}=\sqrt{80}=4 \sqrt{5}$

![](https://cdn.mathpix.com/cropped/2024_04_17_bf0efe45fea0def43891g-1.jpg?height=268&width=437&top_left_y=1285&top_left_x=1301)