Use the transformation rule r_((90^(∘),O))(x,y)= (- y,x) to find the image of a point for a 90^(∘) rotation about the origin O.
Table:
Rotation About
Vertices
The Origin
L'(1,2) M'(2,6) N'(- 2,4)
The Vertex L
L'(2,- 1) M'(3,3) N'(- 1,1)
The Vertex M
L'(5,- 6) M'(6,- 2) N'(2,- 4)
The Vertex N
L'(7,0) M'(8,4) N'(4,2)
Practice makes perfect
Let's draw △ LMN with vertices L(2,- 1), M(6,- 2), and N(4,2).
Then, we will rotate the vertices of the triangle 90^(∘) about the origin and about each vertex of △ LMN.
Rotation About the Origin
Let's use the rule for a 90^(∘) rotation about the origin O to find the images of each vertex of the triangle.
Point
(- y,x)
r_((90^(∘),O))(x,y)=(- y,x)
L( 2, - 1)
(- ( - 1), 2)
L'(1,2)
M( 6, - 2)
(- ( - 2), 6)
M'(2,6)
N( 4, 2)
(- 2, 4)
N'(- 2,4)
Next, we will plot the points and connect them to graph △ L'M'N'.
Rotation About the Vertex L
To find the coordinates of the vertices of the image of △ LMN, we will use the graph of it. To find the image of M, we will draw a 90 ^(∘) angle with vertex L and side LM. Then, we will draw LM' with length LM. We are able to locate N' in the same way.
We have shown the image of △ LMN. The vertices of △ L'M'N' are as follows.
r_((90^(∘),L))(L(2,- 1)) &= L'(2,- 1)
r_((90^(∘),L))(M(6,- 2)) &= M'(3,3)
r_((90^(∘),L))(N(4,2)) & = N'(- 1,1)
Rotation About the Vertex M
We will now find the images of the vertices of △ LMN for a 90^(∘) rotation about M. We will follow the same steps.
We have shown the image of △ LMN. The vertices of △ L'M'N' are as follows.
r_((90^(∘),M))(L(2,- 1)) &= L'(5,- 6)
r_((90^(∘),M))(M(6,- 2)) &= M'(6,- 2)
r_((90^(∘),M))(N(4,2)) & = N'(2,- 4)
Rotation About the Vertex N
Finally, we will rotate △ LMN 90^(∘) about N.
We have shown the image of △ LMN. The vertices of △ L'M'N' are as follows.
r_((90^(∘),M))(L(2,- 1)) &= L'(7,0)
r_((90^(∘),M))(M(6,- 2)) &= M'(8,4)
r_((90^(∘),M))(N(4,2)) & = N'(4,2)
Let's make a table showing the coordinates of the vertices after each rotation.
