Notice that ∠ R'S'Q' is a right angle. Therefore, if these triangles are similar, then R'Q' and RS must be corresponding sides as they are both hypotenuse in their respective triangles.
Orientation
To make sure the triangles have the same orientation, we can rotate △ R'S'Q' by 180^(∘) about point S'.
Position
Since R'' and S are corresponding vertices we can, by translating △ Q''S''R'' 2 units down, map these vertices onto each other. To make the graph a bit clearer, we will remove the red triangle between the midpoints of △ RQS.
Dilation
To map △ Q'''S'''R''' onto △ RQS we have to dilate this triangle by some scale factor. To understand what scale factor we should use, we have to measure the length of two corresponding sides such R'''S''' and SQ. To make the graph a bit clearer, we will remove the green triangle.
The length of S'''R''' is 3 units and the corresponding side SQ is 6 units. This means we have to use a scale factor of 63=2 when we dilate. Notice that we will dilate △ Q'''S'''R''' using R''' as center of dilation.
Since we can map △ QRS onto △ R'Q'S' we know these triangles are similar.
