A(2,1) , B(4,3) and C(3,5) are the vertices of a triangle ABC. If r1 represents the reflection in X axis and r2 represents the reflection in the line x=5, find the image of triangle ABC under combined transformation r2 o r1. Should we perform r2 first or r1?

I first changed the vertices into reflection in the line x=5 and then reflection in x axis. Is that what we are supposed to do?

a reflection in the x-axis can be written as

r1: (x,y) ---> (x, -y)
so:
A(2,1) ---> A1(2,-1)
B(4,3) ---> B2(4,-3)
C(3,5) ---> C3(3,-5)

r2: (x,y) --->(5-x+5,y) = (10-x,y)
A2(2,-1) ---> A3(8,-1)
B2(4,-3) ---> B3(6,-3)
C2(3,-5) ---> C3(7,-5)

suppose we do r2 first
let's look at the first point A
A(2,1) --- (8,1)
and now we will do r1
(8,1) ---> (8,-1)

>b>In this case of transformation, the order in which you do the transformation does not matter, since each one applies only to one of the variables.
Also take a look how your text or instructor has defined r2 o r1
I think most define it as "r2 follows r1", so you would do r1 first, then r2, in a general case

let's look at what happens to (x,y)

if r1, then r2:
(x,y) --->(x,-y) ---> (10-x,-y)
if r2, then r1:
(x,y) ---> (10-x,y) ---> (10-x, -y)

notice we get the same result
As I stated above this will not always happen.

To find the image of triangle ABC under the combined transformation r2 o r1, we need to first determine the order in which the transformations should be applied.

Since the notation "r2 o r1" indicates that the reflection in line x=5 (r2) should be performed first, followed by the reflection in the x-axis (r1), you have correctly identified that r2 should be performed before r1.

To perform the reflection in line x=5 (r2), we need to mirror each point of the triangle across the vertical line x=5. To do this, we subtract the x-coordinate of each point from the line's x-coordinate, and then add the result to the line's x-coordinate.

The image of point A(2,1) under r2 is obtained by subtracting 2 from 5 (x-coordinate of the line) and adding the result to 5. So, the image of A is A'(8,1).

Similarly, the image of point B(4,3) under r2 is B'(6,3), and the image of point C(3,5) under r2 is C'(7,5).

Now, we perform the reflection in the x-axis (r1) on these new points. To do this, we change the sign of their y-coordinates while keeping their x-coordinates the same.

The image of A' under r1 is A''(8,-1).

Similarly, the image of B' under r1 is B''(6,-3), and the image of C' under r1 is C''(7,-5).

Therefore, the image of triangle ABC under the combined transformation r2 o r1 is the triangle with vertices A''(8,-1), B''(6,-3), and C''(7,-5).