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?

Question

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?

Answer 1

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.