Explore relationships between arcs and chords in a circle

1. 9th Grade Honors Geometry
Explore relationships between arcs
and chords in a circle

2. Chord – a segment whose endpoints are on a given circle
Chord of an arc – the chord whose endpoints define the arc
Minor arc – an arc of a circle with measure less than 180 degrees
Major arc – an arc of a circle with measure greater than 180 degrees
Central angle of a chord – the central angle of a circle whose ray intersects
the endpoints of the chord
Central angle of a circle – an angle whose vertex is the center of the circle
Inscribed angle in a circle – an angle with its vertex on the circle and with its
sides each intersecting the circle at a point other than the vertex
Review:

3. https://www.khanacademy.org/math/geometry/cc-geometry-circles/circles/v/language-and-
notation-of-the-circle
Click on the following link for a video from Khan Academy for a formal definition of a circle
and tangent and secant lines.
Video includes diameters, radii, major and minor arcs.

4. Theorem 1: The diameter of a circle is twice its radius.
1. Fold circle A into fourths, and then unfold it.
2. Find and label the diameter and radius on the same fold of the circle.
3. Draw a line on the fold for the diameter in one color and the radius along the
same fold in another color.
4. Label points on the endpoints of the diameter, "D" and "E," so that it's easier
to refer to later.
5. Notice the relationship between the length of the radius and the length of the
diameter. The length of the radius is half the length of the diameter or the
length of the diameter is twice the length of the radius.
6. Justify this relationship using what happens when the circle is folded into
fourths, the diameter is folded in half, and each half is a radius so two radii
make a diameter.

5. Theorem 2: Measure of an arc is related to the measure of the central angle. The sum of a circle's central
angles is 360°.
 Fold a different diameter of Circle A and label the endpoints F and G.
 Label the third diameter (formed by the folding of the circle into fourths).
 See the illustration below as an example of a possible set-up:
 Label some of the central angles. Recall that a central angle is an angle formed at the center of a circle
by two radii. The example above could use angles DAF, DAI, DAE, FAI, and FAE.
 Label some overlapping and non-overlapping central angles and point out the difference. (Example:
angle DAF and angle FAI are non-overlapping, and angle FAI and angle FAE are overlapping.)
 Label the intercepted arcs (example: arc DG).

6.  Note the sum of the measures of a set of non-overlapping central angles is. [eg.m(angle DAF) + m(angle
FAI) = 90°.]
 Note the sum of all the non-overlapping central angles is. [360°.]
 The measure of the arc that goes all the way around the circle is also 360°.
 Note the sum of the measures of the central angles beginning at one endpoint of a diameter and ending
at the other is [180°]
 Note the measure of that intercepted arc is half of the "whole arc," and therefore it measures 180°.
 Note the measure of the central angle formed by the two perpendicular diameters you created when
folding the circle into fourths is [90°], and
 Note the corresponding intercepted arc measures 90° (it is ¼ of the "whole arc").
 Note the relationship between each central angle and its intercepted arc are the same.

7. Theorem 3: Endpoints of a chord are endpoints of an arc and the diameter perpendicular to the chord
bisects the chord and the arc.
 Fold in one edge of circle B to create a chord, but do NOT put the crease through the center this time.
 Label the ends of the chord M and N and label a point on the larger arc P.
 Every chord also creates an arc that is named by the same endpoints as the chord.
 Label the chord (a segment) and the corresponding arc and notice they use the same endpoints.
 Color the minor arc blue and the major arc red and label each.
 In the diagram below, the major arc is referenced by arc MPN and the minor arc is referenced by MN.

8.  Make a diameter perpendicular to chord MN.
 Some examples of a diameter perpendicular to MN are:
 Label the circle's center at point B where the diameters intersect.
 Label point Q at the intersection of the chord and one of the diameters.
 Fold the circle to form a diameter such that point M "touches" point N.
 How does MQ compare to QN?" [The segments are the same length.]
 Note that when chord MN is perpendicular to the diameter, points M and N are equidistant from the
diameter.
 Label point R on the diagram where the perpendicular diameter intersects the arc defined by chord MN.
 Note that a perpendicular diameter bisects the arc. [When we fold the circle to make the perpendicular
diameter, points M and N are equidistant from point Q along the segment MN. By symmetry, points M
and N also are equidistant from point R along the arc. Therefore, the arc also is bisected by the
diameter.]

9. Theorem 4: Measure of an inscribed angle is half the measure of the intercepted arc and half the measure
of the central angle.
 An inscribed angle is an angle in which the vertex is a point on the circle, and the rays of the angle
intersect points on the circle.
 In Circle C fold the circle to form a chord that does not intersect the center of the circle.
 Label the chord with one colored pencil.
 Draw and label three inscribed angles, each with a different colored pencil.
 One ray of each angle should intersect one endpoint of the chord; the other ray of each angle should
intersect the other endpoint of the chord.
 An example of a completed circle is shown below:

10.  Use a protractor to find the measure of each inscribed angle.
 "What do you notice about the angle measures?" [They are all the same measure.]
 Notice the relationship between the central angle and the intercepted arc, they are the same measure.
 Create a central angle using points A, C, and B to determine the length of arc AB.
 "How does the measure of arc AB compare to the measures of the inscribed angles?" [The measure of
the arc is twice that of each angle.]