Undo both actions, starting with the second and ending with the first.
(2, 4), ( 4, 1), (1, 1)
Practice makes perfect
We start with some triangle. Next, we rotate the triangle by 90^(∘) counterclockwise about the origin. Then, we translate its image 1 unit left and 2 units down. After the two transformations, we get a triangle with the following coordinates.
(-5,0), (-2, 2), (-2, -1)
We can find the coordinates of the original triangle by undoing this sequence of actions. Let's think of a real-world example to see how this works. Let's say we want to wear socks and shoes. First we put on our socks and then we put on our shoes.
Later in the day we want to go back to having bare feet. This time we take off the shoes and then we take off the socks.
In the case of our triangle, we need to do something similar. We need to undo the actions that we did starting with the second and ending with the first. Therefore, we should start by reversing the translation.
Reversing the Translation
We know that the second transformation we did to our triangle was a translation by 1 unit left and 2 units down. We can reverse this translation by translating it in the opposite directions by the same numbers of units. Instead of going 1 unit left, we should go 1 unit right. Instead of going 2 units down, we should go 2 units up. Let's apply this new translation to our triangle.
(x, y)
(x + 1, y + 2)
Simplify
( -5, 0)
( -5 + 1, 0 + 2)
(-4, 2)
( -2, 2)
( -2 + 1, 2 + 2)
(-1, 4)
( -2, -1)
( -2 + 1, -1 + 2)
(-1, 1)
This leaves us with the following vertices.
(-4, 2), (-1, 4), (-1, 1)
Next, let's undo the rotation.
Undoing the Rotation
The first transformation we did was a 90^(∘) counterclockwise rotation about the origin. A rotation about the origin by the same number of degrees, but in the opposite direction, will undo the transformation. This means a 90^(∘) clockwise rotation. This rotation changes the coordinates of points in the following way.
ccc
Original Coordinate && Image Coordinate [0.5em]
( x, y) & ⟶ & ( y, - x)
Let's apply it to the points we found after reversing the translation.
(x, y)
( y,- x)
Simplify
( -4, 2)
( 2, - ( -4) )
(2,4)
( -1, 4)
( 4, - ( -1))
(4,1)
( -1, 1)
( 1, -( -1))
(1,1)
Finally, we have the coordinates of the original triangle.
(2, 4), ( 4, 1), (1, 1)
