Intermath
Title
Constructing A Square
Problem Statement
Construct a square with GSP using as many different methods as you can. Describe
the steps involved in each construction.
Explain the steps you would take to construct a regular polygon in Geometer's
Sketchpad. Is there more than one method? Explain.
Problem setup
I am trying to construct a square using different strategies.
Plans to Solve/Investigate the Problem
I plan on playing with the construction tools in GSP to construct squares. I will use the
information I know about a square: all sides are equal in length, all angles equal 90°, and
all sides are perpendicular to each other.
Investigation/Exploration of the Problem:
I.
One way to construct a rectangle…
1. I constructed AB . I then constructed a perpendicular line through vertex
A. I constructed a circle by center + radius around vertex A and
constructed intersection C.
C
A
B
A
B
2. I constructed a perpendicular line through vertex C and vertex B. I hid
A and the perpendicular lines.
C
D
A
C
D
A
B
B

Measurements:
m AC = 3.52 cm mACD = 90.00
m CD = 3.52 cm mCDB = 90.00
m BD = 3.52 cm mDBA = 90.00
m AB = 3.52 cm mBAC = 90.00
II.
A second way to construct a square:
1. I constructed AB . I rotated AB around vertex B. I then rotated BC around
vertex C. With three lines of the square, I constructed a segment to
connect vertex D and A.
A
A
B
B
A
C
B
D
A
C
Measurements:
m AB = 2.70 cm
m BC = 2.70 cm
mDAB = 90.00
mABC = 90.00
m CD = 2.70 cm
mBCD = 90.00
m AD = 2.70 cm
mCDA = 90.00
D
B
C
III.
A third way to construct a square…
1. I constructed A and a line through center A. I then constructed a
perpendicular line through center A. I constructed intersections, vertices
C, D, and E. I connected vertices to form BC , CE , ED , BD . The
segments formed a rectangle within the circle.
B
A
C
B
A
D
E
C
B
A
E
D
C
Measurements:
m BC = 2.83 cm
B
A
F
D
E
m CE = 2.83 cm
mBCE = 90.00
mCED = 90.00
m ED = 2.83 cm
mEDF = 90.00
m BD = 2.83 cm
mDFC = 90.00
A fourth way to construct a square…
1. I constructed AB and then rotated it 90° around vertex B. I constructed a
perpendicular line through vertex A and then another through A’ and line
A. I then hid the perpendicular lines.
IV.
A
B
A'
A
A
B
B
A'
A
B
C
A'
Measurements:
m A'C = 2.41 cm
mABA' = 90.00
mBA'C = 90.00
m BA' = 2.41 cm
mCAB = 90.00
m CA = 2.41 cm
mABA' = 90.00
m AB = 2.41 cm
A
C
B
A'
V.
A fifth way to construct a square:
1. Knowing that a grid has exact measurements, I plotted points that are the
same distance apart from each other: (0,0), (0,5), (5,0), and (5,5).
B
C
A
5 D
4
2
-5
Measurements:
m AB = 5.00 cm
-2
m CB = 5.00 cm
mABC = 90.00
mBCD = 90.00
m CD = 5.00 cm
mCDA = 90.00
m DA = 5.00 cm
mDAB = 90.00
-4
-6
Explain the steps you would take to construct a regular polygon in Geometer's
Sketchpad. Is there more than one method? Explain.
First of all, I realized that a polygon is regular if all its sides are equal and all its angles
are equal. There are multiple methods to construct a regular polygon, as seen above
when I constructed a square using multiple methods. Many methods also exist because
different shapes have different angle and side measurements. For instance, a triangle is
not going to have 3 90° angles like a square has 4 90° angles. So, the first step I would
take is to know regular polygons, the sides and angles that are included in the shape. The
second step I would take is to play around with the tools to construct equal sided and
equal angled figures.
Extension of the problem:
How many ways can you construct regular polygons using GSP?
I.
One way to construct an equilateral triangle:
1. I constructed A . I then constructed a circle by center + point around
vertex B to construct B . I constructed the intersections, deleting one
intersection to leave vertices A, B, and C. I connected these segments to
form an equilateral triangle.
A
A
B
A
A
B
C
B
A
B
Measurements:
II.
m BA = 1.28 cm
mACB = 60.00
m AC = 1.28 cm
mCBA = 60.00
m CB = 1.28 cm
mBAC = 60.00
Another way to construct an equilateral triangle:
1. An equilateral triangle can be constructed within another equilateral
triangle. I constructed an equilateral triangle using the method above. I
constructed vertex E and D as midpoints on BC ' and C ' A . I connected
the segments to form a smaller equilateral triangle within a larger
equilateral triangle.
A
A
A
F
C
C
E
B
F
C
E
B
C
E
B
E
Measurements:
m CF = 1.16 cm
m FE = 1.16 cm
m EC = 1.16 cm
As shown, there is more than one way to construct an equilateral triangle, but the original
triangle must be constructed first.
Author and Contact:
Lauren Mofield, Middle Grades Cohort
Darlnlulu4@yahoo.com