# TRANSFORMATIONS AND CONGRUENCE

Transformations and congruence :

To translate or reflect or rotate a figure in the coordinate plane, we have to transform each of its vertices. Then, we have to connect the vertices to form the image.

When we apply the above mentioned transformations (Translation, Reflection and Rotation), original figure and the image after transformation would have the same size, , just a different orientation. So, the original figure and the image after translation would be congruent.

## Transformations and congruence - Example

A triangle has the vertices (3, 4), (5, 4) and (5, 2).  Apply the indicated series of transformations to the triangle. Each transformation is applied to the image of the previous transformation, not the original figure. Label each image with the letter of the transformation applied.

(i)  Reflect across the x-axis.

(ii)  Translate 3 units to the left.

(iii)  Reflect across the y-axis.

(iv)  Translate 4 units up.

(v)  Rotate of 90 ° clockwise  about the origin.

Compare the size and shape of the final image to that of the original figure.

Solution :

Step 1 :

(i)  Reflect across the x-axis.

Since there is a reflection across the x-axis, we have to multiply each y-coordinate by -1. That is,

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

So, we have

(3, 4) -----> (3, -4)

(5, 4) -----> (5, -4)

(5, 2) -----> (5, -2)

Graph the image.

Step 2 :

(ii)  Translate 3 units to the left.

Since there is a translation of 3 units to the left, we have to subtract 3 from each x-coordinate. That is,

(x, y) -----> (x-3, y)

So, we have

(3, -4) -----> (0, -4)

(5, -4) -----> (2, -4)

(5, -2) -----> (2, -2)

Graph the image.

Step 3 :

(iii)  Reflect across the y-axis.

Since there is a reflection across the y-axis, we have to multiply each x-coordinate by -1. That is,

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

So, we have

(0, -4) -----> (0, -4)

(2, -4) -----> (-2, -4)

(2, -2) -----> (-2, -2)

Graph the image.

Step 4 :

(iv)  Translate 4 units up.

Since there is a translation of 4 units up, we have to add 4 to each y-coordinate. That is,

(x, y) -----> (x, y+4)

So, we have

(0, -4) -----> (0, 0)

(-2, -4) -----> (-2, 0)

(-2, -2) -----> (-2, 2)

Graph the image.

Step 5 :

(v)  Rotate of 90 ° clockwise  about the origin.

Since there is a rotation of 90° clockwise about the origin, we have multiply each x-coordinate by -1 and interchange x and y coordinates. That is,

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

So, we have

(0, 0) -----> (0, 0)

(-2, 0) -----> (0, 2)

(-2, 2) -----> (2, 2)

Graph the image.

Compare the size and shape of the final image to that of the original figure.

They have the same size and shape, just a different orientation.

## Reflect

1. Which transformations change the orientation of figures ?

Reflections and rotations

2.  Which transformations do not change the orientation of figures ?

Translations

3.  Two figures have the same size and shape. What does this indicate about the figures ?

One figure is the image of the other, and there is a sequence of transformations that will transform one figure into the other.

After having gone through the stuff given above, we hope that the students would have understood "Transformations and congruence"

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6