Geometer S Sketchpad Circles Exploration Name

Geometer’s Sketchpad Circles Exploration Name:

Directions: Open Geometer’s Sketchpad by going to the start menu, choosing All Programs, and then going to “The Geometer’s Sketchpad”. Maximize both your windows. From within sketchpad, select “File” then “Open” and navigate to Mrs. Bauman’s student share drive: I:\EFHS\Bauman, Heather (BaumaH)\Geometer's Sketchpad\CirclesExploration.

Central Angles and Inscribed Angles

Objective: By the end of this exploration I will know the relationship between the measure of an arc, a central and an inscribed angle.

B 1. Name the minor arc shown in the drawing : 2. Name the central angle shown: C 3. Name the inscribed angle shown: 4. Conjecture: How do you think the central angle and the measure of the arc are D related? I think that ______. 5. Conjecture: How do you think the inscribed angle and the measure of the arc E are related? I think that ______.

6. Look at the directions on the page titled “Central/Inscribed Angle”. Measure the arcs and angles to fill in the following table (make sure you move point E, C, and B as well as the circle control point to explore what happens).

mBC Central Angle Inscribed Angle

7. Does changing the radius change the measure of an arc? ______8. Does changing the central angle change the measure of an arc? ______9. Does changing where the inscribed angle is in the circle change the measure of the arc (ie, when you moved point E, did that change the arc measure)? ______10. Complete the following statements:

The measure of the arc is ______as the central angle. The measure of the arc is ______of the inscribed angle. The measure of the inscribed angle is ______of the arc measure. The measure of the inscribed angle is ______of the central angle with the same endpoints. Chords

Objective: By the end of this exploration I will know the relationship between two chords of equal measure and the related arc measure.

1. Change to the Chords page in the GSP document. 2. Name two chords in the diagram to the left: and C D 3. Name the two related arcs: and 4. Measure the segments and the arcs in GSP to fill in the following table: CD B EF m C D m EF Chord Chord Arc Measure Arc Measure

5. What do you notice about CD and EF for each of your rows? ______

m C D m EF 6. What do you notice about and for each of your rows? ______

7. If you know that , what would be ______? ______8. If you drew and , what kind of triangle is ∆BCD? How do you know? ∆BCD is ______because ______.

Complete the following statements: If two chords have the same length then ______

If two arcs have the same measure then the chords that connect their endpoints are ______.

If two chords have the same measure, then the central angle formed by their endpoints and the center of the circle are ______. If two central angles are congruent then they define two congruent ______and ______. Tangents Objective: By the end of this exploration I will know the angle formed by the radius and a tangent line and the relationship of two tangent segments when drawn from the same point to the same circle. 1. Name two tangent lines shown in L the picture to the left: J ______and ______

2. Go to the page titled “Single Tangent” and measure . What A kind of angle is it? M P ______K Drag point A, C and B around and O to make sure it holds true. 3. Given that and are radii, find:

N and .

4. Go to the page titled “Two Tangents” and measure and . 5. Make a table and find at least 5 different values when you move around points A, B, and D. BD AD

6. What do you notice about and ? ______

7. If you were to draw triangle ABD, what kind of triangle would it be?

∆ABD is a ______triangle because ______.

8. Name at least two pairs of congruent segments in the diagram above:

______and ______

The angle formed by a radius and a line tangent to the circle at the end of the radius is a ______. If two tangent lines are drawn from the same point outside the circle, then the two line segments formed by that point and the points of tangency are ______.