A 90^(∘) counterclockwise rotation about the origin changes point (x, y) into point (- y,x).
R''(2,1), S''(-1,7), T''(2,7)
Practice makes perfect
Our triangle has the following vertices.
R(-7, -5), S(-1, -2), T(-1, -5)
We first rotate this triangle by 90^(∘) counterclockwise about the origin and then we translate it 3 units left and 8 units up. We want to find the coordinates of the vertices after these transformations. Let's consider each transformation one at a time.
Rotation
When a point is rotated 90^(∘) counterclockwise about the origin, the coordinates always change in the same way.
ccc
Original Coordinate && Image Coordinate [0.5em]
( x, y) & → & ( - y, x)
We can find the coordinates of the rotated triangle using this rule with the given points.
(x, y)
(- y, x)
Simplify
R( -7, -5)
R'( -( -5), -7)
R'(5, -7)
S( -1, -2)
S'( -( -2), -1)
S'(2, -1)
T( -1, -5)
T'( -( -5), -1)
T'(5, -1)
We can also draw the original triangle and then rotate it to get these coordinates. Let's take a look!
Translation
After the rotation, we translate the figure 3 units left and 8 units up. In this transformation the x-coordinate decreases by 3 and the y-coordinate increases by 8.
ccc
Original Coordinate && Image Coordinate [0.5em]
( x, y) & → & ( x - 3, y + 8)
Let's apply this transformation to the vertices of the rotated triangle.
(x, y)
(x-3, y+8)
Simplify
R'( 5, -7)
R''( 5 - 3, -7 + 8)
R''(2,1)
S'( 2, -1)
S''( 2 - 3, -1 + 8)
S''(-1,7)
T'( 5, -1)
T''( 5 - 3, -1 + 8)
T''(2,7)
The triangle we get after the two transformations has the following coordinates.
R''(2,1), S''(-1,7), T''(2,7)
Once again, we can also get these points by showing the transformations graphically. Let's see how the two transformations change the original triangle.
