Page 131 - IM_FL_Geometry_Print Sample

• One partner should hold their pencil vertically with its tip on point while the other partner does the same at point (doing this helps make sure you don’t slide the index card too far in the next step).
• Slide the corner of an index card snugly in between the pencils. This corner makes roughly a right angle.
• Without moving the index card, one of the partners lifts their pencil and marks a point where the corner of the index card is.
• Connect this new marked point to and to create a right triangle.
1. Create at least 10 right triangles that have segment as their hypotenuse.
2. Using tracing paper, trace segment along with all of the new points you marked. What do you notice?
Student Response
1.
2. The vertices at the right angles seem to be tracing out a circle.
Activity Synthesis
Display several tracing papers on top of each other to show that they form a circle. Explain that the legs of the right triangles form what are called chords on the circle. Explain that a chord is a line segment between two points on a circle. A diameter is a special kind of chord that goes through the center of the circle, but other chords don’t have to go through the center.
Ask students to reverse their experiment: construct a circle and diameter with straightedge and compass and choose any point on the circle. Draw chords from that point to the endpoints of the diameter and use the index card to verify experimentally that the angle formed is a right angle. Display one of these constructions and highlight that the central angle forms a straight line on the diameter, making it 180 degrees or radians, and that the right angle is exactly half. Tell students that, while they were able to prove this with coordinates in the previous unit, they will explore a more general connection between central angles and angles formed between chords without coordinates.
6.3 Inscribed Angles vs. Central Angles
15 minutes
58
Teacher Guide