How TR = a/2 in the rectangle? 
I am confused a bit about a theory. 
How $TR= a/2$? is this because of parallel line? 
 A: Yes its because of parallel lines $RQ$ and $OP$.
Hint: Basic Proportionality theorem which states that a line drawn parallel to one side of a triangle divides the other two sides proportionally.
I hope its clear now?
A: A suggestion to you, always try to write-down the given conditions in the question. 
Here, you are given with : 
OPQR is a Trapezoid . What is trapezoid? It is a quadrilateral having two opposite sides as parallel and the other two sides as non-parallel. So, what can you conclude from this? You can simply conclude that it is given that RQ and OP are two parallel sides. 
Now, consider the coordinates given of the points. You are given that : $$\begin{align} & Q \equiv \left(a,a \right) \\ & P \equiv \left(2a,a\right) \end{align} $$
Now, what do you have from the above coordinates? Clearly, the length of $OT \ \text{is} \ a$ and $QT = a$ and $ PT = 2a $ 
Now, try to consider the triangle $\bf{\triangle TOP }$ . In this triangle you have 
$\bf{RQ \parallel OP }$ 
So, from BPT theorem (Basic Proportionality Theorem) , you have : 
$$\begin{align} & \cfrac{TR}{OT} = \cfrac{QT}{PT} \end{align} $$
You have QT = a, PT = 2a and OT = a
Therefore,
$$\begin{align} & \cfrac{TR}{a} = \cfrac{a}{2a} \\ & \require{cancel}\cfrac{TR}{a} = \cfrac{\cancel{a}}{\cancel{2a}} \\ & \boxed{TR = \cfrac{a}{2} }\end{align}$$
A: Yes, it is because of a parallel pair of lines - you know the lines $OP$ and $RQ$ are parallel because $OPQR$ is described as a trapezoid (and the other pair of opposite sides is clearly not parallel).
You should then be able to use the parallels to show that $TRQ$ and $TOP$ are similar triangles.
