NC–PIMS Geometry 6-12
Day 2
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midquad theorem
four similar triangles in a triangle
dilations and isometries
bitmap vs. vector graphics
GSP transformations, iteration and custom tools
Block 1: The Midquad Theorem
D2.B1.A1 Opposite Midsegments in a Quadrilateral
In quadrilateral ABCD, each diagonal separates the quadrilateral into two triangles. The
midsegments of the two triangles must be parallel to the diagonal and half the length of
the diagonal (hence equal to each other). What does that mean about the midpoint
quadrilateral?
A
A
D
D
B
B
C
C
A
D
B
C
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D2.B1.A2
General Midpoint Quadrilateral on a Grid
The activity sheet shows a quadrilateral on a grid with general coordinates. Find the
coordinates of the midpoints of the sides of the quadrilateral. Find the lengths of the
segments joining the midpoints. Find the slopes of the lines joining the midpoints. What
can you conclude?
See the GSP file “Midquad Theorem.gsp” for a demonstration.
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Block 2: Dilations
Halving Triangles into Fourths
D2.B2.A1
On GSP, create an arbitrary scalene triangle. Using each vertex sequentially, dilate the
triangle to a ratio of ½ of the original. Discuss with your partner what you observe. Fill
in the blanks: Dilations preserve a shape’s ____________ but not its __________.
Instructor Notes
Construct arbitrary scalene triangle ABC with midpoints D, E, and F.
Select point A and Transform/Mark Center. Select all vertices and sides of
triangle ABC and use Transform/Dilate and set ratio to ½. Do this again for
points B and C as center. The following figure should be produced.
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The instructor should allow the discussion to develop. Particularly important are
the following concepts:
Dilations create similar triangles. So, each internal triangle is similar to
the original external triangle.
All the internal triangles are congruent.
The area of each internal triangle is ¼ the area of the original.
The discussion in this activity provides students with an opportunity to justify
observations. This is tantamount to informal proof. This experience is valuable
for them.
The following questions should be posed to the students.
Your triangle now has 4 smaller triangles. What can you state about the 4
triangles? How do you know? Can you justify what you state?
Compare each of the small triangles to the original large triangle. How
many relationships between the small triangles and the large
triangles can you find? What can you state about these
relationships? How do you know? Can you justify what you
state?
Discuss different ways of investigating the previous issues and questions.
Could these questions be investigated with manipulatives? How? Could
these questions be investigated through coordinates? How?
Additionally, it should be noted, or demonstrated, that the same investigation can
be performed by paper folding.
GSP Sliders
We have seen how to dilate objects in GSP. Now we will see how to control the dilation
factor with a slider. Our slider will be a segment with a point on it.
Instructor Notes
Demonstrate marking a ratio in GSP. In the diagram below select points C, D, E,
in that order, then select Mark Ratio from the Transform menu. When dilating,
select the “by marked ratio” option.
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D2.B2.A2
Construct a GSP sketch that looks like the picture below. The segment A’ to B’ is a
dilation of segment AB, the center is O and the scale factor is the ratio CE/CD. As you
move the point E the dilation changes.
A
A'
O
B'
B
CE
C
E
D
CD
= 0.60
How does the ratio A’B’/AB compare with CE/CD?
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D2.B2.A3
Create the sketch shown below. How does the ratio of areas X’Y’Z’/XYZ compare with
CE/CD?
X
X'
Y
Y'
O
Z'
Z
CE
C
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D
NC-PIMS Geometry 6-12
CD
= 0.60
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Block 3: Isometries
A dilation is one type of transformation or mapping of the plane (that is, a function that
takes points to points). We have seen that dilations do not preserve length.
An isometry is a transformation that does preserve length. In other words, when two
points X and Y are transformed (mapped) to points X′ and Y′, respectively, then
XY=X′Y′. The word comes from iso, meaning “the same” and metric, meaning
“measure,” so they have the same measure, or same distance. You can think of a
segment XY being moved to segment X 'Y ' , where the two segments are the same
length, so isometries are sometimes called motions.
An isometry:
XX'
Y Y'
XY  X'Y'
Y
X'
X
Y'
There are three basic isometries:
The first isometry is called a translation (or sometimes a slide). It is defined by a vector
AB , where each point P is moved in the direction and distance defined by the vector.
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Translation (slide) defined by vector AB [ PP' = AB ]
A
P
B
P'
The second isometry is called a rotation. The rotation has a center point C and an angle
of rotation (a). Each point Q is rotated about the center C until it makes an angle of a°.
Rotation about center C with angle  [ mQCQ' =  ]
C

Q
Q'
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The third isometry is called a reflection. A reflection has a line of reflection DE , which
is the perpendicular bisector of the point and its image.
Reflection about line DE [ DE is  bis of RR' ]
D
R
R'
E
If you have a geometric figure, like a triangle or a quadrilateral, an isometry will move
the figure to another place, but the image will be congruent to the original. You can also
use a sequence of isometries, because each isometry creates an image congruent to all of
the previous ones.
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D2.B3.A1 Sequence of Isometries
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D2.B3.A2 Transformed Triangles
Suppose you use isometries to “move” a triangle around. How many different figures
can you make. What kind of congruent segments, angles, and figures can you find?
Block 4: Curves – Cutting Corners
Most computer drawing programs have tools for producing smooth curves. Here is a
curve drawn with the Microsoft Paint program.
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Paint saves pictures as bitmap graphics which means that it stores information about a
rectangular grid of pixels. In the picture above, each pixel is either black or white. Some
programs store pictures in a vector graphics format which means that they store
information on how to reproduce the actual picture. For example, if the picture contains
a circle then the vector graphics approach is to record the location of the center and the
radius.
The figure below shows two curves drawn with a program named WinFIG. The curve on
the left is similar to the Paint curve above but it shows the four control points that
determine the shape of the curve. The curve on the right was made by editing the left
curve, dragging control point 2 to the right and point 3 to the left. The vector graphics
versions of these curves require only the locations of the four control points. When the
picture is rendered, the locations of the control points are used to reproduce the actual
curve.
Question: How do you use the control points to produce the curve?
In this section will look at two techniques for going from control points to smooth curves.
Both of these algorithms will be implemented in GSP and those exercises will introduce
some advanced features of the software.
Chaikin’s Corner Cutting Algorithm
The simplest way to go from a set of control points to a curve is to connect consecutive
points with straight line segments forming a piecewise linear curve called the control
polygon. But we want the curve to be smooth and the piecewise linear curve has corners.
Solution: cut the corners off! Find the ¼ and ¾ points on each segment of the control
polygon and connect the ¾ point on one segment to the ¼ point on the next segment to
create a new control
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polygon.
Of course, the new control polygon still has corners, but they are not as sharp. Compare
the angle at B with the angle at C in the figure above. The angle at C is wider, and this
will always be the case. Why? If we iterate this procedure, that is do the same corner
cutting on the second control polygon that we did on the first, and continue, then the
successive control polygons will get closer and closer to a smooth curve.
This corner cutting technique is due to George Chaikin who proved that the control
polygons do approach a smooth curve (An algorithm for high speed curve generation,
Computer Graphics and Image Processing 3 (1974), 346-349).
D2.B4.A1
We are going to implement Chaikin’s algorithm in GSP and learn about two advanced
features of GSP: iteration and custom tools. Start with three control points labeled A, B,
and C as in the figure below.
B
C
A
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Mark center A then dilate point B by ¼ and also by ¾. Now mark center B and dilate
point C by ¼ and by ¾. Construct the segment B’’ to C’.
B
B''
B'
C'
C''
C
A
This is the basic operation that we want to iterate. The steps performed on points A, B
and C need to be performed on B’, B’’ and C’ as well as points B’’, C’ and C’’.
Select points A, B and C then from the Transform menu select Iterate.
In the box next to A you want B’. You can either click the point B’ while the box is
active, or type B’ in the box. B’’ should be beside B and C’ should be beside C. We also
have to iterate the original process over points B’’, C’ and C’’. To do this click the
Structure button in the Iterate dialog box and select “Add New Map”.
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Now put B’’ beside A, C’ beside B and C’’ beside C and click the Iterate button in the
Iterate dialog box.. Your sketch should look like the figure below.
B
B''
B'
C'
C''
C
A
GSP allows you to turn a construction into a custom tool which repeats the steps of the
construction. We will create a custom tool for the corner cutting algorithm.
Select everything in your sketch by using the pointer and dragging a box around the
entire sketch. Now click the double arrow button at the bottom of the GSP tool bar on
the left side and select “Create New Tool”. Name the new tool “Chaikin”. Now when
the double arrow button is active in the tool bar, you are using the custom Chaikin tool.
Use it to create a sketch like the one below.
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Home Work
1. We have looked at the MidQuad Theorem from a number of perspectives
(synthetically, analytically, and through constructions). You have two readings in
your materials which discuss the Van Hiele Levels of geometric understanding
(Linking Theory and Practice in Teaching Geometry, 2005; The van Hiele Levels
of Geometric Understanding). Read the two articles and use the information to
develop a mini-lesson explaining/using the midquad theorem for your students.
Justify what you develop in respect to where you believe your students are in the
Van Hiele levels and how you plan to help them progress to the next levels.
Explain how you would make accommodations for students at different levels.
2. Use GSP to investigate the relationship between the area of a quadrilateral and the
area of its midpoint parallelogram.
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