# TRANSFORMATIONS ON THE COORDINATE PLANE

## About the topic "Transformations on the coordinate plane"

Transformations on the coordinate plane :

We can perform transformations on a coordinate plane by changing the coordinates of the points on a figure. The points on the translated figure are indicated by the prime "symbol" to distinguish them from the original points

## Reflection

A point on a coordinate plane can be reflected across an axis. The reflection is located on the opposite side of the axis, at the same distance from the axis.

## Translation

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  It is a direct isometry.

## Dilation

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.  NOT an isometry.  Forms similar figures.

## Rotation

To rotate a figure 90 degree clockwise about the origin, switch the coordinates of each point and then multiply the new first coordinate by -1.

To rotate 180 degree about origin, multiply both coordinates of each point by -1.

## Reflection on coordinate plane Example 1 :

Graph (3, −2). Then fold your coordinate plane along the y-axis and find the reflection of (3, −2). Record the coordinates of the new point in the table.

Solution : Let us see the next example on "Transformations on the coordinate plane".

Example 2 :

Graph (3, −2). Then fold your coordinate plane along the x-axis and find the reflection of (3, −2). Record the coordinates of the new point in the table.

Solution :   ## Reflection about the line y=x Once students understand the rules which they have to apply for reflection transformation, they can easily make reflection -transformation of a figure.

For example, if we are going to make reflection transformation of the point (2,3) about x-axis, after transformation, the point would be (2,-3). Here the rule we have applied is (x, y) ------> (x, -y).

So we get (2,3) -------> (2,-3).

## Translation on coordinate plane

Translation of h, k :

( x , y ) -----------> ( x + h , y + k )

Translation for (h ,  k) = (-1 , 2)

Let us see the next example on "Transformations on the coordinate plane".

Example 3 :

Let A ( -2, 1), B (2, 4) and (4, 2) be the three vertices of a triangle. If this triangle is translated for ( h, k ) = ( 2, 3) what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is translated for (h , k ) = ( 2 , 3 ). So the rule that we have to apply here is (x , y) -------> ( x + h , y + k )

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the translated triangle A'B'C'

Step 4 :

(x , y) ----------> ( x + h , y + k )

A ( -2, 1 ) ------------ A' ( 0, 4 )

B ( 2, 4 ) ------------ B' ( 4, 7 )

C ( 4, 2 ) ------------ C' ( 6, 5 )

Step 5 :

Vertices of the translated triangle are

A' ( 0, 4) , B ( 4, 7 ) and C' ( 6, 5)

## Dilation on the coordinate plane

Dilation rule

Dilation of scale factor "k" :

( x , y ) -----------> (kx , ky )

Dilation for " k = 2 " Example 4 :

Let A ( -2, -2), B ( -1, 2) and (2 , 1) be the three vertices of a triangle. If this triangle is dilated for the scale factor " k  = 2 ", what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is dilated for the scale factor "k  = 2". So the rule that we have to apply here is (x , y) -------> ( kx , ky )

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the dilated triangle A'B'C'

Step 4 :

(x , y) ----------> ( kx , ky )

A ( -2, -2 ) ------------ A' ( -4, -4 )

B ( -1, 2 ) ------------ B' ( -2, 4 )

C ( 2, 1 ) ------------ C' ( 4, 2 )

Step 5 :

Vertices of the dilated triangle are

A' ( -4, -4) , B ( -2, 4 ) and C' ( 4, 2)

## Rules on finding rotated image ## 90° rotation (clock wise) ## 90° rotation (counter clock wise) ## 180° rotation (clock wise and counter clock wise)

Example 5 :

Let A ( -2, 1), B (2, 4) and C (4, 2) be the three vertices of a triangle. If this triangle is rotated about 90° clockwise, what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is rotated about 90° clock wise. So the rule that we have to apply here is (x , y) -------> (y , -x)

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the reflected triangle A'B'C'

Step 4 :

(x , y) ----------> (y , -x)

A ( -2, 1 ) ------------ A' ( 1, 2 )

B ( 2, 4 ) ------------ B' ( 4, -2 )

C ( 4, 2 ) ------------ C' ( 2, -4 )

Step 5 :

Vertices of the reflected triangle are

A' ( 1, 2) , B ( 4, -2 ) and C' ( 2, -4)

After having gone through the stuff given above, we hope that the students would have understood "Transformations on the coordinate plane".