EXPLORING CHORDS. Q1. Draw and label a radius on the circle. How does a chord compare with a radius? What are their similarities and differences?

Transcription

1 EXPLORING CHORDS Name: Date: In this activity you will be using Geogebra to explore some properties associated with chords within a circle. Please answer each question throughout the activity marked Q# thoughtfully and show work when necessary. What is a chord? Formally, a chord is a line segment that joins two points on the circumference of a circle. In the picture below there is an example of a chord. Q1. Draw and label a radius on the circle. How does a chord compare with a radius? What are their similarities and differences? 1. Construct a circle in Geogebra using the circle tool. 2. Construct a chord by using the segment tool. 3. Construct a radius by using the segment tool. Click on the center of the circle and then on the circle itself. Make sure the radius you have constructed intersects the chord. 4. Construct intersection point between chord and radius by using the Intersect Two Objects tool and clicking on the radius and chord. Q2. Is there a relationship between the radius and the chord you have constructed? Explore this by measuring angles and segment lengths by using the measure tool.

2 5. Construct the midpoint of the chord by selecting the Midpoint of Center tool and then selecting the chord. 6. Drag the radius until it overlaps the midpoint of the chord, and measure angles DGA, CGB (from diagram below) and segment lengths DG and CG. You may want to zoom in to see if you can get closer using the zoom tool. Q3. What happens when the radius intersects the midpoint? Explain this by using angle measures and segment lengths. You may use notation from the corresponding diagram. 7. Is this true for any chord? Construct another circle and a chord. 8. Construct the midpoint of the chord. 9. Use the line tool in order to create the radius so that it always intersects the chord at the midpoint. Do this by selecting the line tool, then choosing the center of the circle and the midpoint of the chord. Measure the angles and segment lengths. Q4. Drag an endpoint of the chord around the circle and write down your observations. What do you see? How can you write what you see as a proposition using proper mathematical terms? Q5. What key ideas did you use today to explore chords in a circle? List at least 3.

3 EXPLORING CHORDS Name: Answer Key Date: In this activity you will be using Geogebra to explore some properties associated with chords within a circle. Please answer each question throughout the activity marked Q# thoughtfully and show work when necessary. What is a chord? Formally, a chord is a line segment that joins two points on the circumference of a circle. In the picture below there is an example of a chord. Q1. Draw and label a radius on the circle. How does a chord compare with a radius? What are their similarities and differences? A chord and a radius have the following similarities: * completely within a circle * have an endpoint on the circle * both segments A chord and a radius have the following differences: * a radius is a constant length * a chord may vary in length between zero and twice the radius * a radius has an endpoint at the center of the circle 5. Construct a circle in Geogebra using the circle tool. 6. Construct a chord by using the segment tool. 7. Construct a radius by using the segment tool. Click on the center of the circle and then on the circle itself. Make sure the radius you have constructed intersects the chord. 8. Construct intersection point between chord and radius by using the Intersect Two Objects tool and clicking on the radius and chord. Q2. Is there a relationship between the radius and the chord you have constructed? Explore this by measuring angles and segment lengths by using the measure tool. I measured the angles created by the intersection of the chord and radius, and also measured the lengths of the segments created by this intersection and haven't noticed any pattern.

4 5. Construct the midpoint of the chord by selecting the Midpoint of Center tool and then selecting the chord. 6. Drag the radius until it overlaps the midpoint of the chord, and measure angles DGA, CGB (from diagram below) and segment lengths DG and CG. You may want to zoom in to see if you can get closer using the zoom tool. Q3. What happens when the radius intersects the midpoint? Explain this by using angle measures and segment lengths. You may use notation from the corresponding diagram. When I drag the point D so that the intersection occurs at the midpoint, I see that all of the angles go to 90 degrees. Since the intersection point is the midpoint of the chord, then the two segments created (GC and DG) are congruent. 7. Is this true for any chord? Construct another circle and a chord. 8. Construct the midpoint of the chord. 9. Use the line tool in order to create the radius so that it always intersects the chord at the midpoint. Do this by selecting the line tool, then choosing the center of the circle and the midpoint of the chord. Measure the angles and segment lengths. Q4. Drag an endpoint of the chord around the circle and write down your observations. What do you see? How can you write what you see as a proposition using proper mathematical terms? Everywhere that I drag the chord I see that the angles are always 90 degrees and since the intersection is at the midpoint, the two segments are congruent. This is called the perpendicular bisector. A proposition for this is that if the radius intersects a chord at its midpoint, then the radius is the perpendicular bisector. Q5. What key ideas did you use today to explore chords in a circle? List at least 3. 1) A radius goes from the center to the circumference 2) A midpoint bisects a segment 3) A perpendicular line meets another line or segment at a 90 degree angle

5 The objective of this activity is to explore a specific proposition that deals with a chord in a circle. A lot of propositions in geometry are lost on the conceptual level since they are presented in a dry manner through a textbook and its only application is for use in a two column proof. I feel that it is important to explore these propositions and try to determine them on our own, so even though this is a simple chord property, it encourages exploration for propositions in the future. I chose to have the students construct a non-specific intersection of a chord and a radius in order to build the proposition from the ground up. I told the students to drag toward the midpoint in order to guide the lesson since this is supposed to be for younger students, but I feel that it is still a good learning experience. I then ask the students to construct a circle that fits the proposition exactly to show that it holds in all cases. I chose to use Geogebra for this lesson because I support open source software in education.