Going in Circles with Sketchpad
Carol A. Marinas,
Ph.D.
Mathematics &
Computer Science
11300 NE 3rd
Court #8
Barry University
Miami, FL 33161
cmarinas@mail.barry.edu
Geometer’s Sketchpad (GSP) has been an
interactive program to create meaning in geometry for a decade. It has mainly been used at the high school
and college level but should be introduced in elementary and middle school.
In particular, middle school students can move from concrete examples of shapes to more abstract representations with Sketchpad. Through guided activities students can learn about the Sketchpad tools, observe geometric relationships, and develop conjectures.
The following guided activities about circles
start with the meaning of a circle (Figure 1) and then progress to
relationships between angles and arcs (Figures 3-6).
Activity #1
1.
Draw a circle
in Geometer’s Sketchpad (GSP).
2.
Draw a segment
from a point on the circle to the center.
Use GSP to determine its length.
3.
Draw 2 more
segments between the center and 2 other points on the circle. Use GSP to determine their length.
4.
Have a class
discussion on the similarities and differences of the results. [The
individual’s segments will be of the same length while each student will have
different lengths from his/her classmates.]
Figure 1 – Radii of a Circle
Terminology: A set
of points equidistant from a center point is a circle.
Terminology: A segment joining the center with a point on
the circle is the radius. How many radii are there for each circle? [infinitely many]
Conjecture: Radii of the same circle are congruent.
Activity #2
1.
Draw a circle
in GSP.
2.
Draw a segment
with endpoints on the circle.
3.
Find the
midpoint of this segment.
4.
Find the
perpendicular through this midpoint.
5.
Draw a second
and third segments with their endpoints on the circle. Repeat steps 3 and 4.
6.
What do these 3
perpendicular lines have in common? [They go through the center of a circle.]
Figure 2 – Chords of a Circle
Terminology: A segment whose endpoints are on the circle
is a chord. What is the longest chord of a circle? [through the center of the circle]
Terminology: A chord through the center of the circle is
the diameter. What is the relationship with the lengths of
a diameter and a radius? [The
diameter is double the radius in length.]
Terminology: A line that is perpendicular and bisects a
segment is called a perpendicular
bisector.
Conjecture: Perpendicular bisectors of chords intersect
at the center of the circle.
Activity #3
1.
Draw a circle
and a radius. Call the center A and the other endpoint B.
2.
Create a line
perpendicular line to the radius at the endpoint B.
3.
Add a point C
on this perpendicular line. Measure the created angle ABC.
4.
Move the
endpoint B around the circle.
5.
What conclusion
can you make? [A perpendicular line to a radius touches the
circle at only one point.]
Figure 3 – Tangents to a Circle
Terminology: A line
that touches a circle at only one point is called a tangent line.
Conjecture: A tangent line is perpendicular to the
radius at the point of tangency.
Activity #4
1.
Draw a
circle.
2.
Create an angle
that has its vertex at the circle’s center.
3.
Measure the
angle.
4.
Measure the
minor arc.
5.
Move the points
and observe the measurements.
6.
What conclusion
can you make? [The measure of the angle whose vertex is the
center of the circle equals the measure of the minor arc.]
Figure 4 – Central Angles of a Circle
Terminology: An angle whose vertex is the center of a
circle is called a central angle.
Conjecture: The measure of a central angle equals the
measure of the intercepted arc.
Activity #5
1.
Draw a
circle.
2.
Create an angle
that has its vertex on the circle.
3.
Measure the
angle.
4.
Measure the minor
arc.
5.
Move the vertex
around the circle.
6.
What conclusion
can you make? [The measure of the angle equals half the
measure of the minor arc.]
7.
What if the
angle intersected the endpoints of a diameter? [The angle is a right angle.]
Figure 5 – Inscribed Angles of a Circle
Terminology: An angle whose vertex is on the circle is
called an inscribed angle.
Conjecture: An inscribed angle has a measure of half the
measure of the intercepted arc.
Conjecture: An angle inscribed in a semicircle has a
measure of 90 degrees.
Activity #6
1.
Draw a
circle.
2.
Draw two chords
that intersection in the interior of the circle.
3.
Measure the
pair of acute vertical angles created at the intersection.
4.
Measure the
intercepted arcs of these vertical angles.
5.
What conclusion
can you make? [The measure of the angle equals half the
measure of the sum of the intercepted arcs.]
Figure 6 – Intersecting Chords in a Circle
Conjecture: An interior angle formed by two intersecting
chords has a measure that is the average of the intercepted arcs.
Activity #7
1.
Draw a
circle.
2.
Draw two rays
whose endpoints intersect in the exterior of the circle.
3.
Measure the
angle created at the intersection.
4.
Measure the
intercepted arcs of this angle.
5.
What conclusion
can you make? [The measure of the angle equals half the
measure of the difference of the intercepted arcs.]
Figure 7 – Rays Intersecting a Circle
Conjecture: An angle whose vertex is in
the exterior of a circle has a measure equal to half the difference of the
intercepted arcs.
Summary
Geometer’s
Sketchpad can be used to:
1.
Define
geometric terms
2.
Create
conjectures about geometric relationships
3.
Use the
interactivity features of GSP to support the conjectures. (moving and animating
points)
4.
Use the table
and calculate options to numerically validate the conjectures
Table Tool
Calculator Tool
Figure 8 – Collection of Data