# 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 ## Examples

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 : 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)

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 of scale factor "k" :

(x, y) -----> (kx, ky)

Dilation for "k  =  2". Example 4 :

Let A(-2, -2), B(-1, 2) and C(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) Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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