8.8 similarity and dilations 1
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This document discusses transformations in geometry, including reflections, similarities, dilations, and their applications. It provides examples of determining if figures are reflections or similar, using proportions to find unknown side lengths, and dilating polygons by multiplying the coordinates by a scale factor. Key points covered are defining similar polygons as those with the same shape but different sizes, using proportions to solve similarity problems, and dilating figures by multiplying the original coordinates by the scale factor.
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Slm understanding quadrilaterals MATHS topic....
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440 Chapter 7 SimilaritySimilar Polygons7-2Objective.docx
440 Chapter 7 Similarity
Similar Polygons7-2
Objective To identify and apply similar polygons
A movie theater screen is in the shape of
a rectangle 45 ft wide by 25 ft high.
Which of the TV screen formats at the
right do you think would show the most
complete scene from a movie shown on the
theater screen? Explain.
Similar fi gures have the same shape but not necessarily the same size. You can
abbreviate is similar to with the symbol ,.
Essential Understanding You can use ratios and proportions to decide whether
two polygons are similar and to fi nd unknown side lengths of similar fi gures.
You write a similarity statement with corresponding vertices in order, just as you write
a congruence statement. When three or more ratios are equal, you can write an
extended proportion. Th e proportion ABGH 5
BC
HI 5
CD
IJ 5
AD
GJ is an extended proportion.
A scale factor is the ratio of corresponding linear measurements
of two similar fi gures. Th e ratio of the lengths of corresponding
sides BC and YZ , or more simply stated, the ratio of
corresponding sides, is BCYZ 5
20
8 5
5
2. So the scale factor
of nABC to nXYZ is 52 or 5 : 2.
Key Concept Similar Polygons
Defi ne
Two polygons are
similar polygons if
corresponding angles are
congruent and if the
lengths of corresponding
sides are proportional.
Diagram
ABCD , GHIJ
Symbols
/A > /G
/B > /H
/C > /I
/D > /J
ABGH 5
BC
HI 5
CD
IJ 5
AD
GJ
CB
A D
IH
G J
C X
Y
Z
B
A
ABC XYZ
15 20
25
6 8
10
Dynamic Activity
Similar Polygons
A
C T I V I T I
E S
D
S
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Lesson
Vocabulary
• similar fi gures
• similar polygons
• extended
proportion
• scale factor
• scale drawing
• scale
L
V
L
V
• s
LL
VVV
• s
a
W
r
c
t
You learned about
ratios in the last
lesson. Can you use
ratios to help you
solve the problem?
hsm11gmse_NA_0702.indd 440 4/15/09 1:49:41 PM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
Problem 2
Got It?
Lesson 7-2 Similar Polygons 441
Understanding Similarity
kMNP , kSRT
A What are the pairs of congruent angles?
/M > /S, /N > /R, and /P > /T
B What is the extended proportion for the ratios of
corresponding sides?
MNSR 5
NP
RT 5
MP
ST
1. DEFG , HJKL.
a. What are the pairs of congruent angles?
b. What is the extended proportion for the ratios of the lengths of
corresponding sides?
Determining Similarity
Are the polygons similar? If they are, write a similarity statement
and give the scale factor.
A JKLM and TUVW
Step 1 Identify pairs of congruent angles.
/J > /T, /K > /U, /L > /V, and /M > /W
Step 2 Compare the ratios of corresponding sides.
JK
TU 5
12
6 5
2
1
KL
UV 5
24
16 5
3
2
LMVW 5
24
14 5
12
7
JM
TW 5
6
6 5
1
1
Corresponding sides are not proportional, so the polygons are not similar.
B kABC and kEFD
Step 1 Identify pairs of congruent angles.
/A > /D, /B > /E , and /C > /F
Step 2 Compare t.
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Dilations 1
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RO Q3 M4 MATH9 pdf.pdf
This document contains a mathematics lesson on solving word problems involving parallelograms, trapezoids, and kites. It begins with an introduction to the key properties of parallelograms, trapezoids, and kites. It then provides 4 examples of word problems involving these shapes and shows the step-by-step work and reasoning to solve each problem using the relevant geometric properties. The lesson aims to teach students how to illustrate, set up, and solve word problems involving parallelograms, trapezoids, and kites.
4.8 congruence transformations and coordinate geometry
1) The document discusses transformations in coordinate geometry including translations, reflections, and rotations. It provides examples of naming the type of transformation based on images and applying transformations to graphs.
2) Examples show applying coordinate rules for translations and reflections to graphs of figures and verifying that the resulting images are congruent.
3) Practice problems at the end review applying transformations like translation notation, reflecting figures, and identifying if a graph is a rotation of another and specifying the angle and direction.
Transformations edmodo 2013
1. The document discusses different types of transformations including translations, reflections, and rotations.
2. A translation slides all points of a figure the same distance in the same direction, preserving congruence. A reflection flips a figure across a line of reflection to create a mirror image, also preserving congruence.
3. Rotations turn all points of a figure about a fixed center point through a given angle, resulting in a congruent figure. Examples are provided of applying transformations to polygons on a coordinate plane.
Similar to 8.8 similarity and dilations 1 (20)
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4.8 congruence transformations and coordinate geometry
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A function is a mathematical rule that assigns each input exactly one output. Functions can be represented as ordered pairs, tables, graphs or equations. When represented as ordered pairs or tables, there cannot be repeats in the x-values, otherwise it is not a function. On a graph, if a vertical line intersects the graph more than once, it is not a function. Functions can be linear or non-linear. A linear function has a constant rate of change, while a non-linear function will have variables with exponents or multiplied together.
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A function is a mathematical rule that assigns each input exactly one output. Functions can be represented as ordered pairs, tables, graphs or equations. When represented as ordered pairs or tables, there cannot be repeats in the x-values, otherwise it is not a function. On a graph, if a vertical line intersects the graph more than once, it is not a function. Functions can be linear or non-linear. A linear function has a constant rate of change, while a non-linear function does not have a straight line graph or variables with exponents or being multiplied.
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8.8 similarity and dilations 1
- 1. Lesson 8.7 , For use with pages 439-444 Tell whether the figure is a reflection in the line shown? 1. 2.
- 2. Lesson 8.7 , For use with pages 439-444 Tell whether the figure is a reflection in the line shown? ANSWER no 1. ANSWER yes 2.
- 3. 8.8 Similarity and Dilations
- 7. EXAMPLE 1 Identifying Similar Polygons STEP 1 Decide whether corresponding angles are congruent. Each angle measures 90 . Tell whether the television screens are similar. A E B F C G D H
- 8. STEP 2 Decide whether corresponding side lengths are proportional. EXAMPLE 1 Identifying Similar Polygons 30 inches 18 inches 40 inches 24 inches = ? 5 3 = 5 3 ANSWER Yes, quadrilateral ABCD ~ quadrilateral EFGH .
- 9. SOLUTION EXAMPLE 2 Standardized Test Practice Corresponding side lengths are proportional. Write a proportion. Substitute given values. Solve the proportion. KL NP = LM PQ KL 12 M = 10M 5 M KL = 24
- 10. EXAMPLE 3 Using Indirect Measurement Height Alma is 5 feet tall and casts a 7 foot shadow. At the same time, a tree casts a 14 foot shadow. The triangles formed are similar. Find the height of the tree.
- 11. SOLUTION EXAMPLE 3 Using Indirect Measurement You can use a proportion to find the height of the tree. Write a proportion. Substitute given values. Solve the proportion. Tree’s height Alma’s height = Length of tree’s shadow Length of Alma’s shadow x feet 5 feet = 14feet 7 feet x 10 = ANSWER The tree is 10 feet tall.
- 17. EXAMPLE 4 Dilating a Polygon Quadrilateral ABCD has vertices A (– 1, – 1), B (0, 1), C (2, 2), and D (3, 0). Dilate using a scale factor of 3 . SOLUTION Original Image (x, y) A (–1, –1 ) B ( 0, 1 ) C ( 2, 2 ) D ( 3, 0 ) ( 3 x , 3 y ) A ’ (– 3, – 3 ) B ’ ( 0, 3 ) C ’ ( 6, 6 ) D ’ ( 9, 0 )
- 18. EXAMPLE 4 Dilating a Polygon Quadrilateral ABCD has vertices A (– 1, – 1), B (0, 1), C (2, 2), and D (3, 0). Dilate using a scale factor of 3 . SOLUTION Graph the quadrilateral. Find the coordinates of the vertices of the image. Graph the image of the quadrilateral. Original Image (x, y) A (–1, –1 ) B ( 0, 1 ) C ( 2, 2 ) D ( 3, 0 ) ( 3 x , 3 y ) A ’ (– 3, – 3 ) B ’ ( 0, 3 ) C ’ ( 6, 6 ) D ’ ( 9, 0 )