2014 Mathcounts State Prep : Folding Paper Questions

Folding paper questions are not too bad so here are two examples:
 

Question #1:  The two side lengths of rectangle ABCD is "a" and "b". If you fold along EF and the point B now converges on point D, what is the length of $\overline{EF}$?

Solution I :
Let the length of $\overline{FC}$  be x and the length of $\overline{FD}$  is thus a - x.

Using Pythagorean theorem you'll easily get x (the $x^2$ part cancel each other out) and from there get the length of a - x.

$\overline{HD}$ = ${\displaystyle \frac{1}{2}}$ of the hypotenuse. (Use Pythagorean theorem or Pythagorean triples to get that length,)

Again, using Pythagorean theorem you'll get the length of $\overline{HF}$. Times 2 to get $\overline{EF}$.

Solution II : 
After you find the length of x, use $\overline{EG}$, which is a - 2x and b as two legs of the right triangle EGF, you can easily get $\overline{EF}$. (Pythagorean theorem)

Question #2: 

What about this time you fold B to touch the other side.
 What is the length of EF? 

This one is not too bad.

Do you see there are two similar triangles?

Just make sure you use the same corresponding sides to get the desired
length.

Answer to one mathleague quesiton from AoPS

Question is here.

You have two congruent triangles. 17-same angle and- x (SAS)
Using distance formula, the two green lines are of the same length.
${(a-25)}^2+{(20-15)}^2=a^2+{20}^2$
a = 5

Use another distance formula to get x -- the blue line.
$\sqrt{{(5-17)}^2+{20}^2}=\sqrt{544}=4\sqrt{34}$