CK-12 Geometry
Page 44
Matrices can be multiplied. Matrix multiplication is a row-by-column operation. Matrix multiplication is not commutative, and not all matrices can even be multiplied.
Multiplication of a polygon matrix by one of the special matrices , , or will reflect the polygon in the axis, the axis, or the line respectively.
Points to Consider
You saw in this lesson that reflections correspond to multiplication by a particular matrix. You might be interested to investigate how multiplication on the right of a polygon matrix by one of the following matrices changes the original matrix.
As if matrix arithmetic is not “different” enough, in an upcoming lesson we’ll see that there is another kind of multiplication called scalar multiplication. Scalar multiplication will enable us to use matrices to represent dilations in the coordinate plane.
Review Questions
Prove that reflection in the line is an isometry.
Given: and , its reflection in
Prove:
Let , and .
Write the matrix for each product.
Let be a polygon matrix. Fill in the blank(s).
If , then __?__.
If , and is a matrix, then is a ___-by___ matrix.
If , then ___?___
If is an matrix, then
Review Answers
Rotation
Learning Objectives
Find the image of a point in a rotation in a coordinate plane.
Recognize that a rotation is an isometry.
Apply matrix multiplication to rotations.
Sample Rotations
In this lesson we limit our study to rotations centered at the origin of a coordinate plane. We begin with some specific examples of rotations. Later we’ll see how these rotations fit into a general formula.
Remember how a rotation is defined. In a rotation centered at the origin with an angle of rotation of a point moves counterclockwise along an arc of a circle. The central angle of the circle measures The original point is one endpoint of the arc, and the image of the original point is the other endpoint of the arc.
180° Rotation
Our first example is rotation through an angle of .
In a rotation, the image of is the point .
Notice:
and are the endpoints of a diameter.
The rotation is the same as a “reflection in the origin.”
A rotation is an isometry. The image of a segment is a congruent segment.
If is a polygon matrix, then the matrix for the image of the polygon in a rotation is the product . The Lesson Exercises include exploration of this matrix for a rotation.
90° Rotation
The next example is a rotation through an angle of .
In a rotation, the image of is the point .
Notice:
and are radii of the same circle, so .
is a right angle.
The acute angle formed by and the axis and the acute angle formed by and the axis are complementary angles.
A rotation is an isometry. The image of a segment is a congruent segment.
If is a polygon matrix, then the matrix for the image of the polygon in a rotation is the product . The Lesson Exercises include exploration of this matrix for a rotation.
Example 1
What are the coordinates of the vertices of in a rotation of ?
Mark axes by s is , is , is
The matrix below represents the vertices of the triangle.
The matrix for the image of is the product:
The vertices of are and
Rotations in General
Let be a polygon matrix. The matrix for the image of the polygon in a rotation of degrees is the product , where
Example 2
Verify that the matrix product for a rotation is a special case of this formula for a general rotation.
We know that and .
For , the general matrix product is
Note that this is the matrix product for a rotation. (See above.)
Example 3
The point is rotated . What are the coordinates of ?
The coordinates of are .
Note: The distance from the origin to is When rotates its image is on the axis, the same distance from the origin as .
Lesson Summary
To find , the image of polygon matrix rotated about the origin:
1. Rotation
2. Rotation
3. Rotation
Points to Consider
You’ve now studied several transformations that are isometries: translations, reflections, and rotations. Yet to come is one more basic transformation that is not an isometry, which is the dilation.
The Lesson Summary above listed a few formulas for rotations. Suppose you only had the first two formulas. Would you be able to find the coordinates of the image of a polygon that rotates or or For that matter, would formula 2 in the summary be enough to find the image of a polygon that rotates These rotations can be solved using compositions of other rotations, a topic coming up in a later lesson.
Review Questions
Let be the point with coordinates .
Write a matrix to represent the coordinates of
Write the matrix for the product
What are the coordinates of the point represented by the product?
Prove that , the origin , and are collinear.
Prove that
A line in the coordinate plane has the equation
Where does the line intersect the axis?
What is the slope of the line?
The line is rotated .
Where does the rotated line intersect the axis?
What is the slope of the rotated line?
What is the equation of the rotated line?
The endpoints of a segment are and . The segment is rotated .
What are the coordinates of and
What is the slope of
What is the slope of
Let . Explain how the product would move point
Review Answers
would not move. The product is equivalent to two rotations, which is equivalent to a rotation, which is equivalent to no rotation.
Composition
Learning Objectives
Understand the meaning of composition.
Plot the image of a point in a composite transformation.
Describe the effect of a composition on a point or polygon.
Supply a single transformation that is equivalent to a composite of two transformations.
Introduction
The word composition comes from Latin roots meaning together, com-, and to put, -position. In this lesson we will “put together” some of the basic isometry transformations: translations, reflections, and rotations. Compositions of these transformations are themselves isometry transformations.
Glide Reflection
A glide reflection is a composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.
The shape below is moved with a glide reflection. It is reflected in the axis, and the image is then translated to the right.
In the diagram, one point is followed to show how it moves. First is reflected in the axis. Its image is . Then the image is translated to the right. The final image is
Example 1
a) What is the image of if it follows the same glide reflection as above?
is reflected in the axis to is translated to the right to . The final image of is .
b) What is the image of ?
is reflected in the axis to is translated to the right to . The final image of is .
Notice that the image of can be found using matrices.
Example 2
How can a rotation of be expressed as a composition?
A rotation is the same as a rotation followed by a rotation. If is the matrix for a polygon, then
is the matrix for the image of in a rotation.
Note that
( rotation
followed by rotation) is also the image of in a rotation.
The triangle below is rotated
Vertices are , and
What are the coordinates of the vertices of the image triangle?
The matrix for the triangle is
The image of the original triangle is shown as the dashed triangle below. The vertices are and .
Vertices of left hand triangle are and vertices of the other triangle are and
Reflections in Two Lines
The Technology Note below gives a preview of how to reflect in two lines.
Technology Note - Geometer's Sketchpad
The following animations show a reflection in two parallel lines, and a reflection in two intersecting lines, in a step-by-step view. Note: Geometer’s Sketchpad software is required to view these files.
At http://tttc.org/find/wpShow.cgi?wpID=1096, scroll down to Downloads. Find:
http://tttc.org/find/wpFile.cgi?id=17534 Composite Reflection Parallel Lines
http://tttc.org/find/wpFile.cgi?id=17532 Composite Reflection Intersecting Lines
Example 3
The star is reflected in the axis. The image of the reflection in the axis is Star'.
Then the image is reflected in the line The image of the reflection of Star' in the line is Star".
One point on the original star, , is tracked as it is moved by the two reflections.
Note that:
Star is right side up, Star is upside down,” and Star is right side up.
is above the axis. is below the axis.
is below the line . is above the line .
Example 4
The trapezoid is reflected in the line . The image of the reflection of Trapezoid in is Trapezoid'.
Then the image is reflected in the axis. The image of the reflection of Trapezoid' in the axis is Trapezoid".
One point on the original arrow box, , is tracked as it is moved by the two reflections.
Note that:
Trapezoid is rotated to produce Trapezoid'.
is above the axis. is below the axis.
Lesson Summary
A composition is a combination of two (or more) transformations that are applied in a specific order. You saw examples of several kinds of compositions.
Glide reflection
Two rotations
Reflection in parallel lines
Reflection in intersecting lines
These compositions are combinations of transformations that are isometries. The compositions are themselves isometries.
For some of the compositions in this lesson you saw a matrix operation that can be used to find the image of a point or polygon. Also, you saw that in some cases there is a simple basic transformation that is equivalent to a composition.
Points to Consider
In this lesson you studied compositions of isometries, which are also isometries. We know that there are other transformations that are not isometries. The prime example is the dilation. We return to dilations in a later lesson, where a second type of multiplication of a matrix is introduced.
Does geometry have a place in art and design? Most people would guess that they do. We’ll get a chance to start to see how when we examine tessellations and symmetry in future lessons.
Review Questions
Explain why the composition of two or more isometries must also be an isometry.
Recall the glide reflection in example 1. Suppose this glide reflection is applied to a triangle, and then applied again to the image of the triangle. Describe how the final image compares to the original triangle.
What one basic transformation is equivalent to a reflection in two parallel lines?
A point is reflected in line . The image is reflected in line . , and the two lines are apart. What is the distance from the original point to the final image?
Point is reflected in two parallel lines. Does it matter in which line is reflected first? Explain.
What one basic transformation is equivalent to a reflection in two perpendicular lines?
Point is reflected in the two axes. Does it matter in which axis is reflected first? Explain.
Prove: Reflection in , followed by reflection in , is equivalent to a rotation.
The glide reflection in example 1 is applied to the “donut” below. It is reflected over the axis, and the image is then translated to the right. The same glide reflection is applied again to the image of the first glide reflection. What are the coordinates of the center of the donut in the final image?
Describe how the final image is related to the original donut.
Review Answers
Images in the first isometry are congruent to the original figure. The same is true of the second isometry. If is a polygon, the image after the first isometry, and the image of after the second isometry, we know that and So i.e., the final image is congruent to the original.
The original triangle has moved to the right.
A translation
Yes. Both final images will be equivalent to translations, but the final images can be different distances from the original point .
A rotation
No. The final image is the same regardless of the order of the two reflections.
Reflection in maps to . Reflection in maps to . Point goes to in the first reflection, then that point goes to in the second reflection. It was established earlier that is equivalent to a rotation.
The original donut has been moved to the right. It’s the same as a translation to the right.
Tessellations
Learning Objectives
Understand the meaning of tessellation.
Determine whether or not a given shape will tessellate.
Identify the regular polygons that will tessellate.
Draw your own tessellation.
Introduction
You’ve seen tessellations before, even if you didn’t call them that!
a tile floor
a brick or block wall
a checker or chess board
a fabric pattern
Here are examples of tessellations, which show two ways that kites can be tessellated:
What is a tessellation? What does it mean to say that a given shape tessellates?
To tessellate with a given shape means that the copies of that shape can cover a plane.
There will be no uncovered gaps.
There will be no overlapping shapes.
The entire plane will be covered, in all directions.
Look at an example.
A quadrilateral fits together just right. If we keep adding more of them, they will entirely cover the plane with no gaps or overlaps. Now the tessellation pattern could be colored creatively to make interesting and/or attractive patterns.
Note: To tessellate, a shape must be able to exactly “surround” a point.
What is the sum of the measures of the angles?
is the sum of the angle measures in any quadrilateral.
Describe which angles come together at a given point to “surround” it.
One of each of the angles of the quadrilateral fit together at each “surrounded” point.
Will all quadrilaterals tessellate? Yes, for the reasons above.
Tessellating with a Regular Polygon
Technology Note- “Virtual” Pattern Blocks-TM
At the National Library of Virtual Manipulatives you can experiment with Pattern Blocks and other shapes. Try to put congruent rhombi, trapezoids, and other quadrilaterals together to make tessellations. The National Library of Virtual Manipulatives can be found at: http://nlvm.usu.edu/en/nav/siteinfo.html.
A good tessellation site is http://nlvm.usu.edu/en/nav/frames_asid_163_g_4_t_3.html?open=activities&from=applets/controller/query/query.htm?qt=tessellations&lang=en.
A square will tessellate. This is obvious, if you’ve ever seen a chessboard or a graph paper grid. A square is a regular polygon. Will all regular polygons tessellate?
Example 2
; Can a regular pentagon tessellate?
Here’s what happens if we try to “surround” a point with congruent regular pentagons.
Regular pentagons can’t surround a point. Three aren’t enough, and four are too many. Remember that each angle of a regular pentagon measures Angles of do not combine to equal the that it takes to surround a point.
Apparently some regular polygons will tessellate and some won’t. You can explore this more in the Lesson Exercises.
Tessellating with Two Regular Polygons
You saw that some regular polygons can tessellate by themselves. If we relax our requirements and allow two regular polygons, more tessellations can be drawn.
Example 3
Here is a tessellation made from squares and regular octagons.