An angle with its vertex at the center of a circle is called a central angle. An angle whose sided are chords of a circle and whose vertex is on the circle is called an inscribed angle. In this activity you will investigate relationships among central angles, inscribed angles, and the arcs they intercept.
1. Construct circle AB.
2. Construct segment AB
3. Construct AC, where C is a point on the circle.
You have just created central angle BAC. Points B and C divide the circle into two arcs. The shorter arc is called a minor arc and the longer arc is called a major arc.
4. Consruct the arc on the circle from point B to point C. You may not see the arc because it overlaps the circle. But you should see its selection indicators.
5. Drag point C around the circle to see how it controls the arc. When you're finished experimenting, locate point C so that arc BC is a minor arc. Click Here
6. Measure the arc angle of BC.
7. Measure angle BAC.
8. Drag point C around the circle again and observe the measures. Pay attention to the differences when the arc is a minor arc and when it is a major arc. Click Here.
Write a conjecture about the measure of the central angle and the measure of the minor arc it intercepts.
Write a conjecture about the measure of the central angle and the measure of the major arc.
9. Construct DC and DB, where point D is a point on the circle, to create inscribed angle CDB.
10. Measure angle CDB.
Click Here to drag points for steps 11-14
11. Drag point C and observe the measures of the arc angle and angle CDB.
Write a conjecture about the measures of an inscribed angle and the arc it intercepts.
12. Drag point D (but not past point C or point B) and observe the measure of angle CDB.
Write a conjecture about all the inscribed angles that intercept the same arc.
13. Drag point C so that the thick arc is as close to being a semicircle as you can make it.
14. Drag point D and observe the measure of angle CDB.
Write a conjecture about angles inscribed in a semicircle.