Session 4, Part C:
Circles (35 mintues)

In This Part: Inscribed Angles | More Circle Constructions

A circle is the set of all points in a plane that are equidistant from a given point in the plane, called the center of the circle. Note 4

Problem C1

Construct at least three circles of different sizes. For each circle, complete steps (a)-(e).

 a. Construct the circle's diameter. (A circle's diameter is the segment that passes through the center and has its endpoints on the circle.) Constructing the diameter of a circle creates two semicircles.

 To construct the diameter of a circle, you can't simply draw a segment that looks like it goes through the center; you need to make sure it's connected to the center of the circle. Here's one way: Change the Segment Tool to a Ray Tool (or Line Tool) by clicking and holding on that button until the other options appear; then select the one you want. Draw a ray or line with one point on the circle and another point connected to the circle's center. (Watch your cursor carefully to make sure it doesn't just look right, but that it really connects to the center.) If you wish, you can draw a segment between the two points where the ray intersects the circle, then hide the ray.   Close Tip To construct the diameter of a circle, you can't simply draw a segment that looks like it goes through the center; you need to make sure it's connected to the center of the circle. Here's one way: Change the Segment Tool to a Ray Tool (or Line Tool) by clicking and holding on that button until the other options appear; then select the one you want. Draw a ray or line with one point on the circle and another point connected to the circle's center. (Watch your cursor carefully to make sure it doesn't just look right, but that it really connects to the center.) If you wish, you can draw a segment between the two points where the ray intersects the circle, then hide the ray.

 b. Construct an inscribed angle in one of the semicircles. (An angle is inscribed in a circle if its vertex is on the circle and its rays intersect the circle. For an angle to be inscribed in a semicircle, the rays must intersect the circle at the endpoints of a diameter.) c. Measure your inscribed angle VXW. d. Grab the vertex of your inscribed angle with the Pointer Tool and move it around the circle. How does this affect the measure of the angle? e. Grab the vertex of your inscribed angle with the Pointer Tool and move it around the circle. How does this affect the measure of the angle?

You can investigate the angles inscribed in semicircles and quarter-circles using Geometer's Sketchpad.

First, investigate the angle inscribed in a semicircle using Problem C1, if you have not done so already.

Problem 2:
To investigate the angle inscribed in a quarter-circle, construct three new circles of different sizes and complete steps (a)-(f).

 a. Construct the circle's diameter to create two semicircles. b. Construct a second diameter in the circle that is perpendicular to the first diameter. This creates four quarter-circles. c. Construct an inscribed angle in one of the quarter-circles. For an angle to be inscribed in a quarter-circle, the rays must intersect the circle at one of the endpoints of each of the diameters. d. Measure your inscribed ACB. e. What conjecture can you make about the measure of an angle that is inscribed in a semicircle? f. What conjecture can you make about the measure of an angle that is inscribed in a quarter-circle?

 Video Segment In this video segment, Ric and Michele construct an inscribed angle in a semicircle, then measure the angle as they move it around the circle. Watch this segment after you have completed Problem C1 and compare your results with those of the onscreen participants. What did Michele and Ric discover about their inscribed angle as they moved it around the semicircle? How does this compare with your findings? If you are using a VCR, you can find this segment on the session video approximately 10 minutes and 50 seconds after the Annenberg Media logo.

 Session 4: Index | Notes | Solutions | Video