Math 803
Unit 8: Transformations and Congruence
8.1 Translations
(6.1 text)
8.2 Reflections
(6.2 text)
8.3 Rotations
(6.3 text)
8.4 Dilations
(6.4 text)
8.5 Transformations
(7.1 text)
8.6 Congruence
(7.2 text)
8.7 Similarity
(7.3 text)
8.8 Similar Polygons
(7.4 text)
(8.G.1, 8.G.2, 8.G.3, 8.G.4)
Name:_____________________________
Period:_________
Teacher’s Name:____________________
803
8.1 Translations
Preimage – the original geometric figure
Image – the NEW figure
Translation – slides a figure into a new position without turning it.
Preimage
Image
Prime symbols - be sure you label the vertices of the new image with prime symbols.
When translating a figure, every point of the pre-image is moved the same
distance and in the same direction.
The image and the pre-image are congruent.
Example:
Graph 𝛥 𝐽𝐾𝐿 with vertices J(-3, 4), K(1, 3), and L(-4, 1). Then graph the image
of 𝛥 𝐽𝐾𝐿 after a translation 2 units right and 5 units down. Write the coordinate
of its vertices.
Practice: Graph 𝛥 𝐴𝐵𝐶 with vertices A(4, -3), B(0, 2) and C(5, 1). Then graph
the image after a translation of 4 units left and 3 units up. Write the coordinates
of the image.
803 8.2: Reflections in the Coordinate Plane (8.G.1)
Reflection: Mirror image of the original figure
Line of Reflection: Line of Symmetry, the line over which the figure is being reflected
803
8.3 Rotations in the Coordinate Plane (8.G.1)
Rotation: a transformation in which a figure is rotated or turned about a fixed
point.
Rotating a Figure About the Origin
803 8.4 Dilations in the Coordinate Plane (8.G.3)
Dilation: a transformation that enlarges or reduces a figure by a scale factor.
Center of dilation: The fixed point at which
Scale Factor = k =
𝑖𝑚𝑎𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
Is the image getting larger or smaller?
To find the vertices of the image, multiply each coordinate of the preimage by the scale factor. (x, y) becomes ________________
803 8.5 Congruence and Transformations (8.G.1, 8.G.2)
Congruent figures: two figures are congruent if the second can be obtained
from the first by a TRANSFORMATION
Determining the Transformations
If you have two congruent figures, you can determine the transformation, or
series of transformations, that maps one figure onto the other by analyzing the
orientation or relative position of the figures.
Translation
•
•
Length is the same
Orientation is the same
Reflection
•
•
A
Length is the same
Orientation is reversed
A
A′
B
B
B′
Rotation
•
•
A′
Length is the same
Orientation is changed
A
A′
B′
B′
B
Ex. 1) Are the two figures below congruent? Explain your reasoning.
A
X
C
B
Z
Y
Ex. 2) Are the two figures below congruent? Explain your reasoning.
What sequences of transformations are performed below?
Mrs. Martinez created the logo shown. What transformations did she use if the
letter “d” is the pre-image and the letter “p” is the image? Are the two figures
congruent?
iamond
lumbing
8.5a Skills Practice
(8.G.1, 8.G.1a, 8.G.1b, 8.G.2)
Congruence and Transformations
Determine if the two figures are congruent by using transformations.
Explain your reasoning.
1.
2.
3.
4.
5.
6.
8.5b Homework Practice
(8.G.1, 8.G.1a, 8.G.1b, 8.G.2)
Congruence and Transformations
Determine if the two figures are congruent by using transformations. Explain your reasoning.
1.
2.
3.
4.
5. GRAPHIC DESIGN The Art Club
designed the logo shown. What
transformations did they use if the top
trapezoid is the preimage and the bottom
trapezoid is the image?
6. SCRAPBOOKING Charlotte used a stamp
to create the pattern shown. What
transformations did she use if
parallelogram A is the preimage and
parallelogram B is the image?
803 8.6 Congruence (8.G.2)
What is Congruence?
Two figures are congruent if…… their corresponding sides and angles are
congruent.
Write the congruence statements comparing the corresponding parts in the
congruent triangles shown.
Corresponding angles:
Corresponding sides:
Quadrilaterals ABCD and PQRS are congruent. Match all the sides and angles.
A) Find m<D
B) m<R
C) Find RS, given that BC is 8; CD is 6.
D) If QR is 4x – 6, what is the value of x?
803 8.6a Skills Practice
(8.G.2)
Congruence
Write congruence statements comparing the corresponding parts in each set of congruent figures.
1.
2.
3.
4.
5.
6.
803 8.7 Similarity and Transformations (8.G.4)
Similar figures have the SAME SHAPE…. BUT DIFFERENT SIZES.
Two figures are similar if the second can be obtained BY TRANSFORMING OR
DILATING the first one.
Example:
Since the orientation of the figures is the same, one of the transformations is a
translation.
𝑘=
𝐴𝐵
𝐷𝐸
6
= =2
3
and
𝑘=
𝐴𝐶
𝐷𝐹
=
10
5
=2
Since the ratios are equal, ΔABC is the dilated image of ΔEDF; therefore the
two figures are similar.
Determine if the two figures are similar by comparing the transformations.
Determine if the two figures are similar by using transformations
An art show offers different size prints of the same painting. The original print
measures 24 centimeters by 30 centimeters. A printer enlarges the original by a
scale factor of 1.5, and then enlarges the second image by a scale factor of 3.
What are the dimensions of the largest print? Are both of the enlarged prints
similar to the original?
803 8.8
Properties of Similar Polygons (8.G.4)
Recall that the ratio of similar figures can be used to solve for
unknown measurements of the figures.
The triangles below are similar. Write a proportion to find the value of x.
Set up a proportion and solve to find the height of the tree.
At the same time a 12-foot adult elephant casts a 4.8-foot shadow, a baby
elephant casts a 2-foot shadow. How tall is the baby elephant?
A surveyor needs to find the distance AB across a pond. He constructs ΔCDE
similar to ΔCAB and measures the distances shown in the figure. Find AB.
803 8.8a
Skills Practice
(8.G.4)
Properties of Similar Polygons
Determine whether each pair of polygons is similar. Explain.
1.
2.
3.
4.
Each pair of polygons is similar. Find each missing side measure.
5.
6.
7.
8.
803
8.8b
Homework Practice
(8.G.4)
Properties of Similar Polygons
Determine whether each pair of polygons is similar. Explain.
1.
2.
Each pair of polygons is similar. Find each missing side measure.
3.
4.
5.
6.
7. TILES A blue rectangular tile and a red rectangular tile are similar. The blue tile has a length of 10 inches
and a perimeter of 30 inches. The red tile has a length of 6 inches. What is the perimeter of the red tile?