Download Regular Polygons

Regular Polygons
[Key Idea 7]
1. Constructing polygons using Tools:
The definition of a regular polygon is an equilateral and equiangular polygon. Equilateral is not
a sufficient condition because rhombi are not regular. Equiangular is not sufficient because
rectangles are not regular. Therefore, to be a regular polygon, the figure must have equal sides
and equal angles.
Identifying the characteristics of regular polygons is an important topic for students, but
constructing them can be tedious. Therefore, Geometer's Sketchpad comes with a file of regular
polygon tools for students to use.
Once in Geometer’s Sketchpad, open the folder labeled "Samples," then the sub-folder labeled
"Custom Tools," and then
the file "polygons.gsp."
You now have a set of tools
that you can use to construct
polygons that can be
accessed through the
Custom Tool button.
Once you select a tool, you
will see at the bottom of the
screen instructions that tell
you what to do to make the
construction.
Schoaff, 2004
Regular Polygons
p. 5
For example, "3/Triangle (By Edge)" needs 2 points.
The instructions will say "1. Match Point A" and then
"2. Match Point B."
If you click on "Show Script View," you can see the
entire script for this construction. This may be useful
for students who wish to learn how to make their own
constructions"
To use this tool, simply select it, move the cursor to the screen, and follow directions.
2. Constructing regular polygons using iterations:
The easiest way to construct a regular n-sided polygon without using a custom tool is by
inscribing the polygon in a circle.
a. Construct a circle. Select the center of the circle, under the Transform menu Mark
Center.
b. Select the point on the circle and under the Transform menu choose Rotate. Enter the
desired angle as 360/n, where n is the number of desired sides to the polygon. (Our
example uses n = 9.)
c. Connect the two points on the circle with a segment.
Schoaff, 2004
Regular Polygons
p. 6
d. Select the first point, and then under the Transform menu choose Iterate. When the
Iterate dialog box comes up, click on the second point. Click on the Iterate button.
e. The program will automatically show several iterations. To complete the polygon,
press the + key.
3. Challenge: There are several methods for constructing equilateral triangles and squares.
How many different ways can you construct an equilateral triangle or a square using the tools
under the Construction and Transformation menus? What properties are you using for each
construction?
Schoaff, 2004
Regular Polygons
p. 7
Triangle Definitions & Properties
[Key Ideas 1, 4, 7]
The definition of a triangle is a 3-sided planar figure. With only this definition, what properties
can we determine?
Exploration #1 Attributes of triangles from three points.
Objective: To construct a triangle from 3 points and explore angle and side measures.
a. Select the Point tool. Make three points by moving to three locations on the screen and
clicking the mouse button. Select the arrow tool to continue.
b. Construct a triangle by selecting all three points. Now under the Construct menu
choose Segment. The points will be connected in the order chosen.
Measure the angles of the triangle. [Select three points, then under the Measure menu
choose Angle. Now unselect the three points by clicking anywhere on the screen or
pressing the esc key.]
Notice that the vertices are now labeled in the order in which they were selected, not
necessarily the order in which they were created.
Repeat selecting three points and measuring for the two remaining angles.
d. Sum the angles of the triangle by choosing Calculate under the Measure menu.
Choose the first measure, click on the +, choose the second measure, +, choose the third
measure, then Okay.]
A
e. Change the size and shape of the
triangle. Point at one of the vertices
(the arrow turns sideways). Hold
down the mouse button and drag the
point on the screen. Release the
button.
m – ABC = 69.66°
m – BCA = 46.08°
m – CAB = 64.26°
Notice that the measurements for the
angles have changed. What stayed the
same?
C
m – A B C + m – B C A + m – CAB = 180.00°
B
f. Measure the three sides of the triangle by selecting all three segments (not the vertices),
and then under the Measure menu choose Length. What do you notice about the
relationship between angle and side measures?
g. What if you drag a vertex so that two angles are equal in measure? What happens to
the sides? What if three sides were equal in length?
What else could you learn from this construction?
Write two possible theorems.
Schoaff, 2004
Regular Polygons
p. 8
Exploration #2 Attributes of a Skewed Triangle
Objective: To construct a triangle with one vertex on a line parallel to the base and then to
determine any special properties.
a. Under the File menu choose New Sketch. This gives you a new screen to work with.
b. Construct the base of the triangle using the Segment tool to draw a segment on the
screen.
c. Construct the third vertex on a line parallel to the base as follows:
- use the Point tool to place a point anywhere on the screen, not on the segment;
- choose the arrow tool and select the point and the line segment;
- then under the Construct menu choose Parallel Line.
The parallel line should now be selected (colored) because it was the last object created.
Under the Construct menu choose Point On Segment. You should now have a
colored point on the parallel line. This is a movable point — practice dragging it.
d. Construct the rest of the triangle. [The movable point on the parallel line is already
selected. Select the two endpoints of the segment . Now under the Construct menu
choose Segment.]
C
m AB = 7.92 cm
Distance C to AB = 3.63 cm
Area
ACB = 14.36 cm2
Perimeter
ACB = 20.45 cm
B
A
e. Measure the length of the base of the triangle by selecting the endpoints. This will
automatically label the base AB . Measure the altitude of the triangle by selecting the
movable point and segment AB, then choose Distance under the Measure menu. This
will automatically label the movable point C.
Measure the area of the triangle as follows: first select points A, B, and C, then under
the Construct menu, choose Triangle Interior. This interior will now be selected
(plaid), so under the Measure menu choose Area.
Measure the perimeter of the triangle. Select the interior, then under the Measure
menu choose Perimeter. (Make sure you deselect the area measure first.)
Schoaff, 2004
Regular Polygons
p. 9
f. Move vertex C along the parallel line by selecting point C, then holding down the
mouse button and dragging it. Since it is a child of the parallel line, it is restricted to
moving along this line. Dragging the vertex of a triangle like this is called skewing the
triangle.
What changes? What stays the same? Why?
Write a conjecture. (A conjecture is something that you think might be true. A conjecture, once
proven, becomes a theorem. Many students list definitions as conjectures, but we don’t ‘prove’
definitions.)
Can you change the area without changing the perimeter? Describe how you might do this.
Exploration #3 Constructing Equilateral Triangle
An equilateral triangle is defined as a triangle with 3 equal sides.
C
m AB = 4.08 cm
m AC = 4.08 cm
B
A
a.
To construct an equilateral
triangle, start with a clean screen.
With the segment tool, draw a
segment.
b.
Select one endpoint and
then select the other endpoint.
Under the Construct menu,
choose Circle by Center+Point.
While the circle is selected, under
the Construct menu, choose
Point on Circle. Construct a
segment from the center of the
circle to the new point. Measure
both segments AB and AC.
What do you notice? Can you explain why?
c. Select point C and delete it.
Select point B and then select point A. Under the Construct menu, choose Circle by
Center+Point. Notice that the two circles intersect in 2 places. Using the arrow, point
to one intersection and click. This places a point at the intersection. Select points A, B,
and the intersection point, then Construct Segments. Measure the three segments to
confirm that we have constructed an equilateral triangle using the definition (3 equal
sides).
Schoaff, 2004
Regular Polygons
p. 10
m AB = 4.08 cm
D
m DA = 4.08 cm
m BD = 4.08 cm
B
A
d. Select the two circles, then under the Display menu choose Hide Circles. Measure the
angles of the triangle. Make a conjecture.
If ∆ABD is equilateral then
Now drag point A or B to determine if your conjecture seems to hold. When you drag, what
changes? What stays the same?
(Note: you cannot drag D to change the size of the triangle because D is a child of the circles,
which are children of points A and B.)
3B. Converse
Is the converse true? That is, if ∆ABD is equiangular, is it also equilateral? (Are having 60˚
angles both necessary and sufficient for an equilateral triangle?)
a. To determine this, start with a new segment.
- Select one endpoint and under the Transform menu choose Mark Center (it will
flash).
- Select the other endpoint and under the
Transform menu choose Rotate and then enter
the angle 60˚.
- Then select the first endpoint and the new point,
in that order. Under Construct choose Ray.
m – BAB' = 60.00°
- Measure the angle to verify that it is 60. Hide to
point on the ray.
b.
Now similarly construct a 60˚ angle at B by
selecting
B and under the Transform menu
B
choose Mark Center. Select point A and under
the Transform menu choose Rotate and then enter the angle –60˚ (because we are rotating
clockwise). Where did A' land?
A
Schoaff, 2004
Regular Polygons
p. 11
c. Construct ray BA'. Measure angle ABA'. Hide the point A'.
d. You should now have a triangle. Construct a point at the intersection of the two rays.
Measure the length of segment AB. Measure the distance from A to the intersection
point. Measure the distance from B to the intersection point. What kind of triangle do
you have?
(Note: you may want to discuss the difference in notation here, mAB as opposed to AB.)
If ∆ABC is equiangular then
Exploration #4 Properties of Medians of Equilateral Triangles
A median in a triangle is a segment that joins the midpoint of a side of a triangle to the opposite
vertex.
a. Construct the medians of one of the equilateral triangles previously constructed. For
example, using equilateral ∆ABC, select all three sides, then under the Construct menu
choose Midpoints. Now connect each midpoint with the opposite vertex.
b. Measure the lengths of the medians. What do you notice?
In an equilateral triangle the medians are _____________________________.
_
C
D
Measure the two angles formed at each vertex by the median.
(For example, –ACF and –BCF in my figure.) What do you
notice?
E
G
In an equilateral triangle, the medians are also ____________
______________________
A
F
B
Measure the two angles formed by the median at the midpoint
of the opposite side. (For example, –BDA and –BDC in my
figure.) What do you notice?
In an equilateral triangle, the medians are also ________________________________
Conclusion:
If ∆ABD is equilateral then the medians
Schoaff, 2004
Regular Polygons
p. 12
Exploration #5 Constructing an Isosceles Triangle
An isosceles triangle is defined as a triangle with two congruent sides.
a. Start with a clear screen and a segment AB. We will use part of the construction for an
equilateral ∆.
C
m AB = 4.90 cm
m CA = 4.90 cm
m BC = 7.74 cm
_b. Select point A and then select point B.
Under the Construct menu, choose Circle by
Center+Point. While the circle is selected,
under the Construct menu, choose Point on
Circle. Construct segment AC. Measure both
segments AB and AC.
c.
Construct segment BC. Measure this
segment. Hide the circle.
A
d.
Measure all three angles. What do you
notice? Make a conjecture about the angles of an
isosceles triangle.
B
If ∆ABC is isosceles then _____________
e.
Can you drag the vertices of the isosceles
∆ABC until all three sides are congruent? If you can, that means that equilateral
triangles are a subset of isosceles triangles. That is, all equilateral triangles are special
isosceles triangles.
f. Notice that in Exploration #3, when you drag a vertex of an equilateral triangle, it is
always an acute triangle because all angles are less than 90˚. Can you drag a vertex of
the isosceles triangle to make it acute?
Can you drag a vertex of the isosceles triangle to make it obtuse?
Can you drag a vertex of the isosceles triangle to make it a right triangle?
Make a conjecture.
If ∆ABC is equilateral, then it can be _____________, it can never be
If ∆ABC is isosceles, then it can be
,
it can never be
5B. Converse: Is the converse true? That is, if a triangle has two equal angles, is it always
isosceles? (Is having two equal angles both necessary and sufficient condition for an isosceles
triangle?)
a. To determine this, start with a new segment DE.
- Select point D and under the Transform menu choose Mark Center.
- Select point E and under the Transform menu choose Rotate and then enter any
angle measure less than 90˚.
Schoaff, 2004
Regular Polygons
p. 13
- Select points D and the resulting point in that order. Under Construct choose Ray.
- Measure the angle.
b. Now similarly construct the same angle at E by selecting E and under the Transform
menu choose Mark Center. Select point D and under the Transform menu choose
Rotate and then enter the negative of the angle measure you used before.
Construct the ray from E through the constructed point. Measure the angle.
F
m – E'DE = 75.00°
m – D'ED = 75.00°
FD = 10.36 cm
FE = 10.36 cm
c.
You should now have a triangle.
Construct a point where the two rays intersect
and label it F. Measure the distance DF and EF.
What kind of triangle do you have?
If ∆DEF has two equal angles then
___________________________________.
m AB = 6.39 cm
m CA = 6.39 cm
C
m BC = 9.01 cm
m DB = 7.13 cm
D
E
m FC = 7.13 cm
Exploration #6 Properties of Medians of
Isosceles Triangles
a. Construct the medians of the isosceles
triangle ABC by selecting all three
sides, then under the Construct menu
choose Midpoints. Now connect
each midpoint with the opposite
vertex. Label the intersection.
m EA = 4.54 cm
D
E
G
A
b. Measure the lengths of the medians.
What do you notice?
B
F
Measure the two angles formed at each vertex by the median. (For example, –CAE,
–BAE, –ACF, –BCF, etc. in my figure.) What do you notice?
Measure the two angles formed by the median at the midpoint of the opposite side. .
(For example, –AEC, –AEB, –BDA, –BDC, etc. in my figure.) What do you notice?
Schoaff, 2004
Regular Polygons
p. 14
If ∆ABC is isosceles then the medians to the equal sides _________________________
If ∆ABC is isosceles then the median to the unequal side ________________________
c. Is there any relationship between the lengths of the segments of the medians? In other
words, using my labels, what is the ratio between AG :GE? The ratio BG :GD? The
ratio CG : GF? Does this appear to be true for other kinds of triangles?
6B. Converse:
d. You should have noticed that if ∆ABC is isosceles, then the median from the unequal
angle (vertex angle) is also an angle bisector and altitude. Is the converse true? In
other words, are any two of these attributes (median, angle bisector, altitude) sufficient
conditions for an isosceles triangle?
That is, if you construct a point on the ^ bisector of the base (median + altitude), will
the triangle always be isosceles?
Similarly, if you had any angle with a point on its angle bisector and you constructed a
line through that point perpendicular to the angle bisector (angle bisector + altitude),
would the resulting triangle be isosceles?
A
D
40.6°
40.6°
90.0°
90.0°
D
90.0° 90.0°
B
C
A
AC = 3.68 cm
BC = 3.68 cm
Obviously, similar explorations can be done with other classifications of triangles, such as
scalene, right, acute, obtuse, and combinations of these.
Schoaff, 2004
Regular Polygons
p. 15
Coordinate Geometry
[Key Idea 1, 3, 4, 5, 7]
Exploration #1 Reflections using Coordinates
Under Edit Preferences, set the unit distance to cm. and units, set the angle to degree and tenths,
and set the computations to tenths. Open a new sketch, from the Graph menu choose Show
Grid and Snap Points. Select the two points at the origin and at (1,0) and under Display choose
Hide Points.
a. Place points on the grid to create an irregular polygon. Select the points in order and
Construct Segments. Select the points and under Measure choose Coordinates.
b. Select the y-axis and under Transform mark as Mirror.
c. Select the entire polygon. Reflect. Measure the new coordinates.
d. Compare the coordinates of the original points and the reflected points. Write a rule for
reflecting in the y-axis.
A': (-3.0, 6.0)
B': (-3.0, 2.0)
C': (-4.0, 2.0)
D': (-4.0, 3.0)
E': (-7.0, 3.0)
F': (-7.0, 4.0)
G': (-5.0, 4.0)
H': (-7.0, 6.0)
-10
H'
A'
F'
A
H
4
G'
E'
6
G
D'
C'
D
B'
2
-5
B
F
E
C
A: (3.0, 6.0)
B: (3.0, 2.0)
C: (4.0, 2.0)
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
H: (7.0, 6.0)
5
10
e. Undo the reflection and its coordinates. Select the x-axis and under Transform mark as
Mirror.
f. Select the original polygon. Reflect. Measure the new coordinates.
g. Compare the coordinates of the original points and the reflected points. Write a rule for
reflecting in the x-axis.
h. Undo the reflection and its coordinates. Plot a point at (0,0) and (1,1). Construct the
line through these points, which is the line y = x. Select this line and under the
Transform menu mark it as a Mirror.
i. Select the original polygon. Reflect. Measure the new coordinates.
j. Compare the coordinates of the original points and the reflected points. Write a rule for
reflecting in the line y = x.
Schoaff, 2004
Regular Polygons
p. 16
k. Undo the reflection and its coordinates. Select the origin and the line y = x. Construct
a perpendicular. This is the line y = –x. Select this line and under the Transform menu
mark it as a Mirror.
6
A
H
4
G
D
2
-10
B
E
C
-5
F
A: (3.0, 6.0)
B: (3.0, 2.0)
C: (4.0, 2.0)
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
H: (7.0, 6.0)
5
10
-2
A': (-6.0, -3.0)
B': (-2.0, -3.0)
C': (-2.0, -4.0)
D': (-3.0, -4.0)
E': (-3.0, -7.0)
F': (-4.0, -7.0)
G': (-4.0, -5.0)
H': (-6.0, -7.0)
A'
B'
D'
C'
-4
G'
-6
H'
F'
E'
-8
l. Select the original polygon. Reflect. Measure the new coordinates.
m. Compare the coordinates of the original points and the reflected points. Write a rule for
reflecting in the line y = –x.
Exploration #2 Translations using Coordinates
a. Use the same set-up as in Exploration #1, including step (a).
b. Select the entire polygon. Under the Transform menu choose Translate. Choose By
Rectangular Vector. Enter a horizontal of –3 and vertical distance of –5. Translate.
c. Measure the new coordinates.
Schoaff, 2004
Regular Polygons
p. 17
d. Compare the coordinates of the
original points and the translated
points. Write a rule for translating
by a rectangular vector.
6
A
H
4
G
F
D
2
B
A'
E
C
H'
G'
F'
D'
B'
E'
C'
-4
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
H: (7.0, 6.0)
A': (0.0, 1.0)
5
-2
A: (3.0, 6.0)
B: (3.0, 2.0)
C: (4.0, 2.0)
B': (0.0, -3.0)
C': (1.0, -3.0)
D': (1.0, -2.0)
E': (4.0, -2.0)
10
F': (4.0, -1.0)
G': (2.0, -1.0)
H': (4.0, 1.0)
Exploration #3 Rotations with Coordinates
a. Choose the origin and mark it as a Center. Use the same set-up as in Exploration #1,
including step (a).
b. Select the entire polygon. Under the Transform menu choose Rotate. Enter an angle of
90˚.
c. Measure the new coordinates.
d. Compare the coordinates of the original points and the rotated points. Write a rule for
rotating by a 90˚.
H': (-6.0, 7.0)
G': (-4.0, 5.0)
F': (-4.0, 7.0)
E': (-3.0, 7.0)
D': (-3.0, 4.0)
C': (-2.0, 4.0)
B': (-2.0, 3.0)
A': (-6.0, 3.0)
H'
F'
E'
6
A
H
G'
D'
A'
C'
4
G
B'
D
2
B
C
F
E
A: (3.0, 6.0)
B: (3.0, 2.0)
C: (4.0, 2.0)
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
H: (7.0, 6.0)
e. Undo the rotation and coordinates of the new shape. Select the original polygon.
Under the Transform menu choose Rotate. Enter an angle of 180˚.
f. Measure the new coordinates.
g. Compare the coordinates of the original points and the rotated points. Write a rule for
rotating by a 180˚.
Schoaff, 2004
Regular Polygons
p. 18
h. Again undo the rotated polygon and its coordinates. Select the original polygon and
6
A
4
G
D
2
B
H
A: (3.0, 6.0)
B: (3.0, 2.0)
F
C: (4.0, 2.0)
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
E
C
-5
H: (7.0, 6.0)
5
10
A': (-3.0, -6.0)
C'
E'
B': (-3.0,
-2.0)
-2
C': (-4.0, -2.0)
D': (-4.0, -3.0)
B'
D'
F'
E': (-7.0,
-3.0)
-4
F': (-7.0, -4.0)
G': (-5.0, -4.0)
H': (-7.0, -6.0)
G'
H'
-6
A'
under the Transform menu choose Rotate. Enter an angle of 270˚.
i. Measure the new coordinates.
j. Compare the coordinates of the original points and the rotated points. Write a rule for
rotating by a 270˚.
6
A
H
4
G
F
D
2
B
E
C
5
A: (3.0, 6.0)
B: (3.0, 2.0)
C: (4.0, 2.0)
D: (4.0, 3.0)
E: (7.0, 3.0)
F: (7.0, 4.0)
G: (5.0, 4.0)
H: (7.0, 6.0)
10
-2
B'
-4
C'
A'
D'
G'
-6
E'
F'
H'
A': (6.0, -3.0)
B': (2.0, -3.0)
C': (2.0, -4.0)
D': (3.0, -4.0)
E': (3.0, -7.0)
F': (4.0, -7.0)
G': (4.0, -5.0)
H': (6.0, -7.0)
-8
Schoaff, 2004
Regular Polygons
p. 19