Properties of Circles
Using GeoGebra, you will be constructing chords and inscribed angles of the circle.
Based on your investigations, you will make conjectures regarding properties and
theorems of the circle.
Chords
Perpendicular Lines from Centre to Chord
Construction:
i.
Construct a circle; label centre and the given point on the circle.
ii.
Construct and label a second point on the circle.
iii.
Construct a line segment between the two points on the circle, forming a
chord.
iv.
Construct a line perpendicular to the chord, which passes through the
centre of the circle; construct and label the point of intersection formed.
Measurements:
i.
What should the measurement of the angles formed between the chord and the
line be? Verify by measuring each angle.
ii.
Measure the distances from the endpoints of the chord to the point of the
intersection with the perpendicular line. What do you notice?
Investigation:
i.
Using the original point on the circle, increase and decrease the size of the
circle. What do you notice about the measurements you made?
ii.
Using the second point on the circle, move it around the circumference,
changing the length of the chord. What do you notice about the measurements
you made?
iii.
Complete the instructions on the conjecture note page.
Congruent Chords
Construction and Measurements:
i.
Construct and label a circle with four additional points; do not use the original
point that was part of the circle’s original construction.
ii.
Construct two different chords; measure both chords.
iii.
Construct the midpoint of each chord.
Investigation:
i.
Move the endpoint of one of the two chords until BOTH chords are congruent
or equal in length.
ii.
Measure the distances from the centre to the midpoints of both chords. What
is true about these measures?
iii.
Next, select the MOVE tool, click on the original point on the circle’s
circumference. Move that point so that a larger circle is formed. What do you
notice about the measurements you have made? Lastly, move the original
point so that a smaller circle is formed. What do you notice about the
measurements you have made?
Inscribed Angles
Congruent Inscribed and Central Angles
Construction:
i.
Construct and label a circle with three additional points on the circle
ii.
Construct a central angle which is subtended by the minor arc formed by the
two constructed points on the circle
iii.
Construct an inscribed angle which uses the original point on the circle as its
vertex, and is subtended by the same minor arc endpoints as the central angle
Measurements:
Measure both the central and inscribed angles. What do you notice?
Investigation:
i.
Use the original point on the circle to change its size; what do you notice
about the measures you made?
ii.
Move either one of the two endpoints to change the sizes of the inscribed and
central angles; keep the measure of the central angle less than 180º. What do
you notice about the measures of the angles?
iii.
Complete the instructions on the conjecture note page.
Congruent Inscribed Angles
Construction:
i.
construct and label a circle and 3 additional points
ii.
Using a minor arc formed by two of the additional points, construct the central
land inscribed angles subtended by the arc, with the third additional point as
the vertex of the inscribed angle.
Measurement:
Measure the central angle, measure the inscribed angle.
Investigation:
i.
Move the vertex of the inscribed angle along the circumference of the circle,
between the two endpoints. What do you notice about the measures?
ii.
Change the sizes of the inscribed and central angles by moving one of the
endpoints, keeping the central angle less than 180º. Again, move the vertex of
the inscribed angle along the circle. What do you notice about the measures?
Inscribed Angles Subtended by a Semicircle
An inscribed angle subtended by a semicircle could also be described as an angle
inscribed in a semicircle or an angle subtended by a diameter of the circle.
i.
What do you know about the measure of this type of inscribed angle? Why?
ii.
Obtain a printout showing a labeled circle, a diameter and two inscribed
angles subtended by the diameter. Include the measures to illustrate your
conjecture above and state the conjecture.
Tangents
Tangent and Radius
Construction
i. ........Construct a circle and one additional point.
ii.
Construct a radius from the center to the additional point.
iii.
Construct a tangent line to the circle so that it touches at the additional point.
iv.
Construct a point somewhere on the tangent line.
Measurement
i
Measure the angle between the radius and the tangent line (use the three
points to identify the angle). What type of angle is formed?
Investigate
i
Using the original point on the circumference of the circle, make the circle
larger and smaller. What do you notice about the measure of the angle
between the Radius and the tangent line?
ii
Using the end point of the radius, move the point along the circle’s
circumference. What do you notice about the measure of the angle between
the radius and the tangent line?
Tangent lines from an exterior point
Construction
i
Construct a circle.
ii
Construct a point in the exterior of the circle.
iii
Construct a tangent line from the exterior point to the circle. Is only one line
drawn?
iv
Construct the points of intersection between the circle and the tangent lines.
Measurement
i
Measure the line segments from the exterior point to each point of intersection
on the circle (called “tangent segments). What do you notice about the
lengths of these segments?
Investigation
i
Move the exterior point so that it is closer to the circle and then further from
the circle; what do you notice about the lengths of the tangent segments?
ii
Move the original point on the circle’s circumference so that the circle
becomes smaller and larger. What do you notice about the lengths of the
tangent segments?
Think about….
What type of triangles are formed if a segment was drawn from the exterior
point to the center of the circle, and radii were drawn from the center to each
point of tangency? Why?