Question

Two right triangles on a coordinate plane are similar but not congruent.Each of the legs of both triangle are extended by 1 unit,creating two new right triangles.Are the resulting triangles similar? explain using an example

Answer 1

Virtually every time you try this stunt, the new triangles
are NOT similar.

Example:

Triangle A₀ . . . legs = 3 and 4;  hypotenuse = √25  = 5 .
Triangle B₀ . . . legs = 6 and 8;  hypotenuse = √100 = 10 .

Ratio of short legs = 6/3 = 2
Ratio of long legs  = 8/4 = 2
Ratio of hypotenuses = 10/5 = 2
The triangles are similar. 

Add 1 unit to all legs:

Triangle A₁ . . .  legs = 4 and 5;  hypotenuse = √41
Triangle B₁ . . .  legs = 7 and 9;  hypotenuse = √130

Ratio of short legs = 7/4 = 1.75
Ratio of long legs  = 9/5 = 1.8
Ratio of hypotenees = √(130/41) = 1.78 .

New triangles are NOT similar.

I think this ONLY works when the original triangles are
congruent, so the ratio of their similarity is 1:1. 
When each leg of both triangles is extended by the same
amount, they are still congruent.
 
But the question stipulates that they are NOT congruent.
So the answer to the question is:   No.  Never.
 

Answer 2

They are similar, by SSS. This shows that the sides are all similar but may not be congruent. And they started out similar, and they have the same scale factor. If you had one more angle besides the 90, you would be able to figure out the angles inside and have AA to justify. 
Sorry if this is a bad explanation.