Grade 9: Transformation Geometry October 2013 Grade 9 Transformation Geometry Goals: □ Recognize, describe, and perform transformations with points, line segments and simple geometric figures on a co-ordinate plane, focusing on: o Translation within and across quadrants. o Reflecting across the y-axis and x-axis. o Reflection across the line y=x □ Identify what the transformation of a point is, if given the co-ordinates of its image. □ Use proportion to describe the effect of enlargement or reduction on area and perimeter of geometric figures. □ Investigate the co-ordinates of the vertices of figures that have been enlarged or reduced by a given scale factor. Terminology Transformation Translation Reflection Enlargement/Reduction Scale Factor Co-ordinate 1
Transcript
Grade 9: Transformation Geometry
October 2013
Grade 9
Transformation GeometryGoals:
□ Recognize, describe, and perform transformations with points, line segments and simple geometric figures on a co-ordinate plane, focusing on:
o Translation within and across quadrants.o Reflecting across the y-axis and x-axis.o Reflection across the line y=x
□ Identify what the transformation of a point is, if given the co-ordinates of its image.□ Use proportion to describe the effect of enlargement or reduction on area and
perimeter of geometric figures.□ Investigate the co-ordinates of the vertices of figures that have been enlarged or
Δ ABC has been translated up and to the right. We call Δ A ' B ' C ' the image of Δ ABC under the transformation.
Δ ABC has been reflected across a line of
Δ ABC has been rotated 90 ° clockwise around point O.
Δ ABC has been enlarged by a scale factor of 2. Note that this is NOT a transformation because Δ ABC is NOT congruent to
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Grade 9: Transformation Geometry
TranslationsIntroduction: On an xy-axis, a transformation occurs whenever a shape is moved (whether by flip, rotation, etc.) so that it has the same measurements at the end as it did at the beginning. There are three common types of transformations:
1) Translations: Moving a shape up and down or side to side.
2) Reflections: Flipping a shape over a line of symmetry.
3) Rotations: Turning a shape from 0 ° to 360 °. We will NOT discuss rotations in the co-ordinate plane in grade 9.
NB – Note that later in this section we will talk about reductions and enlargements. These are not transformations because the segments change size.
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Line of Symmetry
Grade 9: Transformation Geometry
Translations.
To translate a point left or right, change the x part of the point’s co-ordinate.To translate a point up or down, change the y part of the point’s co-ordinate.
To translate an entire shape, simply translate each of the points individually. Consider the following example:
e.g. 1 - Translate the triangle XYZ 6 units to the right, giving the co-ordinates of the new triangle X ' Y ' Z '
X=(−4 ;4 );Y=(−2 ;3 );Z=(−4 ;1)
Answer – Increase all of the x-values by 6. Hence the co-ordinates of Δ X 'Y ' Z ' are:
X=(2; 4 );Y=(4 ;3 );Z=(2 ;1)
This answer is confirmed by the graph at right.
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To translate the point A right 2 units, increase its x-value by 2.
A=(−4 ;2 )→A '=(−2;2)
To translate the point B up 3 units, increase its y-value by 3.
B=(1 ;1 )→B'=(1 ;4 )
To translate the point D down 2 units, decrease its y-value by 2.
D= (−2;−2 )→D'=(−2 ;−4)
To translate the point C left 4 units, decrease its x-value by 4 .
C=(5 ;−2 )→C'=¿
Grade 9: Transformation Geometry
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Grade 9: Transformation Geometry
Exercise 1
Match each drawing to its correct description.
Answer Name A B
1.1 Enlargement
1.2 Reflection
1.3 Translation C D
1.4 Rotation
1.
2. Give the co-ordinates of the following points. The first example is done for you.
Description Answer
e.g. Point A translated up 2 A'=(2 ;5 )
2.1 Point B translated left 4 B'=()
2.2 Point C translated up 3 and left 2
C '=()
2.3 Point D translated right 2 and down 1
D'=()
2.4 Point E translated left 10 and down 3
E'=()
2.5 Point F translated up 9 and right 7
F '=()
3. The triangle KLM has co-ordinates K=(2 ;2), L=(0 ;−1), M=(−1 ;2). 3.1. Graph the co-ordinates, then translate the triangle down 3 units and right 2 units to
make triangle K ' L' M '. 3.2. Give the co-ordinates of triangle K ' L' M '.
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Grade 9: Transformation Geometry
4. The point D=(2 ;−1) has been translated 4 different ways to the points D1, D2, D3, and D4. Match each point to its correct description.
Point Translations
4.1 Up 1, Left 2
4.2 Up 5, Left 6
4.3 Down 1, Right 2
4.4 Down 1, Left 5
5. Stephanie walks 300 meters north of her house. She then walks 300 meters west, followed by 100 meters south, then 100 meters east, then finally 200 meters south. How far is she from her house? Draw a graph or picture to support your answer.
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Grade 9: Transformation Geometry
ReflectionsA co-ordinate (or shape) can be reflected over any line of symmetry. In particular, we’ll look at reflections over the x-axis, the y-axis, and the line y=x .
Reflection across the x-axis: To reflect a point over the x-axis, change the sign of the point’s y-value. For example, the point A=(5 ;2) becomes A1=(5 ;−2) in the example below.
Reflection across the y-axis: To reflect a point over the y-axis, change the sign of the point’s x-value. For example, the point A=(5 ;2) becomes A2=(−5 ;2) in the example below.
Reflection across the line y=x : To reflect a point over the line y=x , switch the point’s x- and y-values. For example, the point A=(5 ;2) becomes A3=(2 ;5) in the example below.
E.g. 1. Given that S=(−2 ;3) and T=(0 ;3), Reflect the segment ST across the x- axis to find the coordinates of segment S1T1. Then reflect S1T1across the line y=x to find the segments S2T2. What is the length of S2T2.
Answer: S1=(−2 ;−3) and T 1=(0;−3). Applying the
second reflection to S1T1, we have that S2=(−3 ;−2) and T 2=(−3; 0). Looking at S2T2 on the graph makes it clear that the length of S2T2 is 2, which is the same length as ST and S1T1.
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Grade 9: Transformation Geometry
Exercise 2
1. Decide if the following transformations are reflections or translations.
1.1 1.1 1.2
1.2
1.3 1.3 1.4
1.4
1.5 1.5 1.6
1.6
1.7 1.7 1.8
1.8
1.2. Give the co-ordinates of the following points. The first example is done for you.
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Description Answer
e.g. Point A reflected about x-axis. A'=(2 ;−4 )
2.1 Point B reflected about x-axis. B'=()
2.2 Point C reflected about y-axis. C '=()
2.3 Point D reflected about y-axis. D'=()
2.4 Point E reflected about y=x . E'=()
2.5 Point F reflected about y=x . F '=()
Grade 9: Transformation Geometry
3. Triangle KLM has co-ordinates K=(−2 ;2), L=(−3 ;3), M=(−3 ;2). 3.1. Draw KLM , then reflect the triangle over the y-axis to make triangle K1 L1M 1. 3.2. Give the co-ordinates of triangle K 1 L1M 1.3.3. Reflect triangle KLM over the x-axis and give the co-ordinates of the triangle
K 2 L2M 2
4. The quadrilateral EFGH has co-ordinates E=(0 ;0), F=(1;1), G=3 ;−1¿, H=(2 ;−2). 4.1. Graph EFGH and reflect it over the line y=x to make quadrilateral E ' F ' G ' H ' . 4.2. Give the co-ordinates of quadrilateral E ' F ' G ' H ' .
5. Decide if each pair of points represent a reflection over the x-axis, the y-axis, or the line y=x .5.1. A and A '5.2. B and B'5.3. C and C '5.4. D and D '
6. For each example, identify which transformation is used. (Note: In some cases, there may be more than one correct answer.) The first example is done for you.e.g. A=(−1;1 )→A '=(1;5 ) B=(4 ;1 )→B'=(6 ;5). Translation up 4 and right 26.1. X=(−3 ;4 )→X '=(4 ;−3).6.2. C=(6 ;0 )→C'=(6 ;0); D= (4 ;2 )→D'=(4 ;−2); E=(7 ;−3 )→E'=(7 ;−3)6.3. Y= (5 ;7 )→Y '=(0 ;4).6.4. F=(0 ;0 )→F '=(0 ;0); G= (−2;0 )→G'=(2 ;0).6.5. W=(7 ;0 )→W '=(7 ;0); Z=(6 ;−2 )→Z '=(6 ;2).6.6. H= (0;−2 )→H '=(−2 ;0).
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Grade 9: Transformation Geometry
Enlargements and ReductionsTo enlarge a shape, multiply each of the co-ordinates values by a scale factor.
For example, to enlarge the following triangle ABC by a scale factor of 2, we’ll multiply each of the co-ordinate values by 2:
A=(1;1 )→ (1×2 ;1×2 ), so A'=(2 ;2)
B=(3 ;1 )→ (3×2 ;1×2 ), so B'=(6 ;2)
C=(1 ;2 )→ (1×2;2×2 ), so C '=(2 ;4 )
The two triangles are shown on the axes at right.
To reduce a shape, divide each of the co-ordinates values by a scale factor.
For example, to reduce the following quadrilateral QUAD by a scale factor of 2, we’ll divide each of the co-ordinate values by 2:
Q= (−4 ; 4 )→ (−4÷2 ;4÷2 ), so Q'=(−2 ;2)
U=(6 ;0 )→ (6÷2 ;0÷2 ), so U '=(3 ;0)
A=(4 ;−4 )→ (4÷2;−4÷2 ), so A'=(2 ;−2)
D= (−2;−4 )→ (−2÷2;−4÷2 ), so D'=(−1;−2)
The two quadrilaterals are shown on the axes at left.
E.g. 1 – Reduce the following triangle XYZ by a factor of 6: X=(15 ;12 );Y=(0 ;9 ) ;Z= (−12;24 ).
X '=( 156 ; 126 )=(2,5 ;2 )
Y '=( 06 ; 96 );(0;1,5)Z'=(−126 ; 24
6 )=(−2; 4)
Triangle X ' Y ' Z ' is shown at right.
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Answer:
Grade 9: Transformation Geometry
Exercise 3
1. Decide if each of the following are reductions or enlargements. Then write down the scale factor
2. The triangle MLK is enlarged by a factor of 2. Given that M=(1 ;1 ); L=(1 ;3) and K= (−4 ;2 ) ...2.1. Find the co-ordinates of M ' L ' K ' .2.2. Plot both MLK and M ' L'K ' on an xy axis.2.3. What kind of triangles are MLK and M ' L'K '?
3. The quadrilateral LAMB is reduced by a factor of 4. Given that L=(8; 4 ); A=(−8; 4); M=(−6 ;−4) and B=(6 ;−4 ) ...3.1. Find the co-ordinates of L ' A ' M ' B ' .3.2. Plot both LAMB and L ' A ' M ' B' on an xy axis.
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Grade 9: Transformation Geometry
3.3. What kind of quadrilaterals are LAMB and L ' A ' M ' B'?
4. Using any co-ordinates you choose, make a triangle and enlarge it by a scale factor of 2. Draw both the triangle and its enlargement on one xy axis.
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Grade 9: Transformation Geometry
Enlargement and Reduction InvestigationExercise 4
1. Consider the square SQAR, with co-ordinates S=(−2 ;−2); Q=(2 ;−2); A=(2 ;2); and R=(−2 ;2):1.1. Draw SQAR on an xy axis.1.2. What are the area and perimeter of SQAR?1.3. Enlarge SQAR by a scale factor of 3 and draw the resulting square, S1Q1 A1 R1, on
the same axes as SQAR.1.4. What are the area and perimeter of S1Q1 A1 R1?1.5. Reduce SQAR by a scale factor of 2 and draw the resulting square, S2Q2 A2 R2, on
the same axes as SQAR.1.6. What are the area and perimeter of S2Q2 A2 R2?
2. Consider triangle ABC, with A=(0 ;6); B=(0 ;0); and C=(8 ;0):
2.1. Draw ABC on an xy axis.2.2. What are the area and perimeter of triangle ABC?2.3. Enlarge ABC by a scale factor of 3 and draw the resulting triangle, A1B1C1, on
the same axes as ABC.2.4. What are the area and perimeter of triangle A1B1C1?2.5. Reduce ABC by a scale factor of 2 and draw the resulting triangle, A2B2C2, on
the same axes as ABC.2.6. What are the area and perimeter of A2B2C2?
3. Suppose a rhombus with an area of 100 is enlarged by a scale factor of 3. What is the area of the enlarged circle?
4. In general, if we enlarge a shape with area A by a scale factor of 3, what is the area of the enlarged shape?
5. Suppose a rectangle with an area of 100 is reduced by a scale factor of 2. What is the area of the reduced rectangle?
6. In general, if we reduce a shape with an area of A by a scale factor of 2, what is the area of the reduced shape?
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Grade 9: Transformation Geometry
Problem Solving
Exercise 5
1. Stephanie walks 300 meters north of her house. She then walks 300 meters west,
followed by 100 meters south, then 100 meters east, then finally 200 meters south. How
far is she from her house? Draw a graph or picture to support your answer.
2. Stephanie’s younger brother decided to walk the same distances, only in the reverse
order. That is, he walked 200 meters south, then 100 meters east, then 100 meters
south, then 300 meters west, and finally 300 meters north. How far is he from his
house? How does this answer compare to question 1?
3. A fixed point is a point which does not change under a transformation. For example, the
point A=¿) is fixed when reflected over the x-axis because A'=A=(3; 0).
3.1. Which points are fixed when reflected over the x-axis?
3.2. Which points are fixed when reflected over the y-axis?
3.3. Which points are fixed when reflected over the line y=x?
3.4. Which single point is fixed under all of the reflections mentioned above?
4. When performing an enlargement, what happens if we use a negative scale factor? For
example, try enlarging the following triangle ABC by a scale factor of −2.
A=(1 ;1) B=(1;2) C=(2;1)
5. When performing an enlargement, what happens if we use a scale factor which is less
than one? For example, try enlarging the following triangle ¿ by a scale factor of 1/2.
D=(2 ;2) E=(2 ;4 ) F=(4 ;2)
6. What other kind of reductions or enlargements (i.e. besides multiplying or dividing co-
ordinates) could you use to change a shape? Explain your idea and draw an example.
7. Consider an enlargement in 3-dimensions. If a cube with a volume of 1 is enlarged by a
scale factor of 2, what is the area of the new cube?
