Transformations Similarity and
Congruence
UMI: July 19, 2016
Transformation
A transformation is a process which
changes the position (and possibly the
size and orientation) of a shape. There
are four types of transformations:
•
•
•
•
Rotation (Turn)
Reflection (Flip)
Translation (Slide)
Enlargement (Resize)
Rotation (Turn)
Rotation (also known as Turn) The distance from
the center to any point on the shape stays the same.
Every point makes a circle around the center.
• Changes the orientation of the shape
• Changes the position of the shape
• Everything else stays the same.
Example:
Rectangle A′B′C′D′ is the image
of rectangle ABCD after which
of the following rotations? Find
the new coordinate of the
A′B′C′D′
A.
B.
C.
D.
A 90° clockwise rotation about the origin
A 180° rotation about the origin
A 90° counterclockwise rotation about the origin
A 90° counterclockwise rotation about the point (1, 1)
Solution:
Solution: C is correct.
By joining A and A′ to the origin O, it is clear that this is a
90° counterclockwise rotation about the origin. The same
distance from the origin.(Do the same for the other points
and their images to see for yourself). The new coordinate
is −1,2 , −3,2 , −3,5 , −1,5
Practice 1:
The square is rotated one
complete turn about the
point O. Which of the
following shows the new
position of the square?
If it's a complete turn, then the square will
finish in the same position that it started, so the
answer is D.
Practice 2:
The square is rotated half a complete
turn about the point O. Which of the
following shows the new position of
the square? Find the new coordinate
of the square.
The point O stays in the same place. But the other corners
of the square go to the opposite side of O. So the answer is
B. The new coordinate is 0,0 , 4,0 , −4， − 4 , 0，
Practice 3:
When this 'L'-shape is rotated about
the origin (0,0) by 90°anticlockwise
(counterclockwise), which one of
these would it look like?
Practice 3:
1. shows a rotation of 90° clockwise about (0,0)
2. shows a rotation of 180° about (0,0)
3. shows a rotation of 90° anticlockwise about
(0,0)
4. shows a rotation of 90° anticlockwise, but
the center of rotation is (1,1)
The correct answer is C
Reflection(Flip)
Reflection (also known as Flip) in a line produces a
mirror image in which corresponding points on the
original shape and the mirror image are always the
same distance from the mirror line.
• Every point is the same distance from the central line !
• The reflection has the same size as the original image
• The central line is called the Mirror Line
How Do I Do It Myself?
1. Measure from 2. Measure the 3. Then connect
the point to the
same distance
the new dots up!
mirror line
again on the
(must hit the
other side and
mirror line at
place a dot.
a right angle)
Practice 1:
The square is flipped across the line
L. Which of the following shows the
new position of the square?
It's like reflecting in a mirror. So the new square must be on the opposite
side of L and the same distance away from L. The correct answer is A.
Practice 2:
Rectangle A′B′C′D′ is
the image of rectangle
ABCD after
reflection in which of
the following lines?
A.
B.
C.
D.
The x axis
The y axis
The line y = x
The line y = -x
Solution:
The correct answer is C. Each point and its image
must be the same distance from the mirror line, which
is y = x
Practice 3:
The rectangle is reflected in the line
y = 4.Which one of the following
shows the correct image?
A shows a reflection in the line x = -5
B shows a reflection in the y axis, or a reflection in the x axis
C shows a reflection in the line y = -3
D correctly shows a reflection in the line y = 4
Translation (Slide)
Translation (also known as Slide) moves a shape by
sliding it up, down, sideways or diagonally, without turning
it or making it bigger or smaller.
• Without rotating, resizing or anything else, just moving
• Every point of the shape must move:
 the same distance
 in the same direction.
Translation can be written down as the coordinates change
𝑥, 𝑦 → 𝑥 + 𝑎, 𝑦 + 𝑏
Example:
Find the new image of
triangle ABC using the
translation:
𝑥, 𝑦 → 𝑥 − 2, 𝑦 + 3)
Solution :
𝐴 1,1 → 𝐴‘ −1,4
𝐵 6,5 → 𝐵‘ 4,8
𝐶 7,1 → 𝐶‘ 5,4
Practice 1:
Write a rule to describe the
translation:
Solution :
Let’s see that: 𝑌 4,1 → 𝑌′ −6, −2
The rule for this translation is
𝑥, 𝑦 → 𝑥 − 10, 𝑦 − 3
Practice 2:
The L-shape
A′B′C′D′E′F′ is the
image of the L-shape
ABCDEF after which
of the following
translations?
A. 5 units in the negative x direction and 8 units in the
negative y direction
B. 8 units in the negative x direction and 5 units in the
negative y direction
C. 8 units in the positive x direction and 5 units in the positive
y direction
D. 5 units in the positive x direction and 8 units in the positive
y direction
Solution:
The correct answer is D. The point A has moved 5
units in the positive x direction and 8 units in the
positive y direction. Similarly, for each of the other
5
points. This is denoted by the translation vector
8
Practice 3:
The 'L'-shape is translated 2 units in
the positive x direction and 4 units in
the positive y direction. Which one of
the following shows the correct image?
C correctly shows a translation of 2 units in the positive x
direction and 4 units in the positive y direction.
Similar
Two shapes are Similar when the only
difference is size (and possibly the
need to move, turn or flip one around).
Similar figures
• have equal corresponding angles
• the length of their corresponding sides have equal
ratios
Example:
Determine whether the
polygons are similar.
Justify your answer
Solution :
1. The corresponding angles are equal.
7
3
7
3
2. 10.5 = 4.5 = 10.5 = 4.5 , the measures of the sides of
the polygons are proportional.
Therefore, the polygons are similar.
Example:
Determine whether the
polygons are similar.
Justify your answer
Solution :
1.
4
3
8
6
4
3
8
,
6
= = = the measures of the sides of the
polygons are proportional.
2. The corresponding angles are not equal.
Therefore, the polygons are not similar.
Example:
The triangles are similar.
Find the value of x
Solution :
Write proportions using corresponding parts. Then solve
to find the missing measure
𝑥 4
=
4 8
Solve 𝑥 = 2
Practice 1:
The two trapezoids are
similar. What is the value
of x?
Solution :
The small trapezoid is a scaled-down version of the
large trapezoid, but is also a mirror image. So the side x
corresponds to the length 4 in the larger trapezoid. Write
proportions using corresponding parts. Then solve to
find the missing measure
𝑥 10
=
4 16
Solve 𝑥 = 2.5
Practice 2:
BC is parallel to DE. What
is the length of BC?
Solution :
Because BC is parallel to DE, pairs of corresponding
angles are equal. So triangles ABC and ADE have the
same angles and are similar triangles. Write proportions
using corresponding parts. Then solve to find the
missing measure
𝐵𝐶 5
=
9.9 9
5
𝐵𝐶 = × 9.9 = 5.5
9
Congruent
If one shape can become another using Turns,
Flips and/or Slides, then the shapes
are Congruent.
Congruent figures
• The same size
• The same shape.
• They are similar figures that are equal in size
Example:
These figures are congruent.
Find the measure of angle I.
Solution :
1. Since the figures are congruent, all angles have the
same measure.
2. According to the figures; m∠𝑄 = 42𝑜 , 𝑠𝑜 𝑚∠𝐼 = 42𝑜
Example:
The two quadrilaterals
are congruent. Which
side in quadrilateral
ABCD corresponds to
WZ in quadrilateral
WXYZ?
Solution :
1. WZ faces the angles marked with two arcs and four arcs.
2. DA also faces the angles marked with two arcs and four
arcs.
3. So DA corresponds to WZ.
Thank You
References:
1. A problem solving approach to mathematics
for elementary school teachers by Billstein,
Libeskind, Lott
2. Math is Fun : https://www.mathsisfun.com/
3. Emathematics: http://www.emathematics.net/