Reflected points are the same distance from the line of reflection but on opposite sides of the line before and after the reflection takes place.
J''(3,- 3), K''(12,- 9), L''(3, - 15)
Practice makes perfect
We are given the vertices of a figure.
J(1,1), K(3,4), L(5,1)
Let's start by plotting the vertices in a coordinate plane and connect them with segments to draw our figure.
We want to rotate and then dilate it using a scale factor of 3. Let's do one transformation at time!
Rotation
A rotation is a transformation about a fixed point called center of rotation. Each point of the original figure and its image are the same distance from the center of rotation. When a clockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.
We want to rotate the given figure 90^(∘) clockwise about the origin. Let's use the information in the table to find the coordinates of the image of each vertex.
ccc
Preimage & & Image
(x,y) & → & (y,- x) [0.5em]
J(1,1) & & J'(1,- 1) [0.5em]
K(3,4) & & K'(4,- 3) [0.5em]
L(5,1) & & L'(1,- 5)
We can now plot the obtained points and draw the image of the given triangle after the rotation!
Dilation
A dilation can be an enlargement or a reduction of the preimage. Which type of dilation it is depends on the value of the scale factor k.
Enlargement
k>1
Reduction
0
When the center of dilation in the coordinate plane is the origin, each coordinate of the preimage is multiplied by the scale factor k to find the coordinates of the image.
ccc
Preimage & & Image [0.5em]
(x,y)& ⇒ & ( kx, ky)
Now, let's find the coordinates of the vertices of J'K'L' after a dilation with a scale factor k= 3.
Dilation With Scale Factor k=3
Preimage
Multiply by k
Image
J'(1,-1)
( 3(1), 3(- 1))
J''(3,- 3)
K'(4,- 3)
( 3(4), 3(- 3))
K''(12,- 9)
L'(1,- 5)
( 3(1), 3(- 5))
L''(3,- 15 )
We can now plot the points and connect them with segments to draw the image.
The final coordinates of the vertices of the transformed are J''(3,- 3), K''(12,- 9), and L''(3, - 15).
