The electric field at the center of a square

[Jay9313]

http://helios.augustana.edu/~dr/203/probs/Set%201.pdf
Scroll down to #11 for a picture.

So I kind of understand the math. I am a little confuse though. So why is a divided by sqrt(2)?
If this is a vector, then the center is really (a(sqrt(2)))/2 and the x component would be divided by sqrt(2) which makes the equation equal a/2. Can someone explain this? Is my math wrong?

 

[Spinnor]

The diagonal distance from corner to corner is a*2^.5 and the distance from corner to the center is half of that,

a*2^.5/2 = a/2^.5

Yes?

 

[Redbelly98]

The distance from the center to any corner is $a/\sqrt{2}$, that's why a is divided by sqrt(2).

Taking the x-component is accounted for by the cos(45) in the expression.

 

[Jay9313]

I'm not sure why it's a/sqrt(2). I do know that (a(sqrt(2))/2 is the exact same number. I don't know why.

Basically, I found the length of the diagonal, a(sqrt(2)) and divided it by 2. It's the same as a/sqrt(2)