The goal of this inscribed angle lab is to discover and explain some important relationships among angles in a circle. Then these relationships will be applied to solve some problems and prove some additional theorems. (This will cover the geometry explained in sections 5.2 and 5.3 of BG.)
Some new terminology will be introduced: inscribed angles, central angles, arcs, arc subtended by an angle.
Draw a circle through A with center O. Then construct the point P as the intersection of the RAY AO with the circle, so that AP is a diameter.
Construct a point B on the circle and then construct segments AB,
Measure angle POB and angle PAB.
Compare the measurements of the two angles and find a relationship that holds true wherever B moves on the circle.
Find a geometrical relationship in the figure and then use them to explain (i.e., prove) that the relationship is true. Specifically, answer these questions:
Write a statement of the theorem that you have found in your notes.
Make a new page in your document by duplicating page 1 (by Document Options > Add Page > Duplicate > 1).
Important: Check preferences and make sure that Angle Measure is still Directed Degrees.
Now use your Angle Metering Tool on O and A to create the same circle but with a new point C on the circle, along with the segments, angle measures, etc.
Compute the differences (angle POB – angle POC) and (angle PAB – angle PAC).
Also measure angle COB and angle CAB. The angle measures and the differences should give the same numbers wherever you move B or C. (Check this.)
Now what is the relationship between angle COB and angle CAB for any locations of B or C? Explain why this is true based on what you found in Investigations 1 and 2.
If c is a circle, an inscribed angle in the circle is defined to be an angle CAB, where points A, B and C are all on the circle.
A central angle of a circle is an angle COB, where B and C are points on the circle and O is the center of the circle.
The Inscribed Angle Theorem says this: If A, B, C are points on a circle c with center O, the inscribed angle CAB equals one-half the central angle COB.
You have proved this theorem in your investigations. But sure that you both understand the theorem and also that you can put your observations together as a proof.
Note: This theorem is stated and proved in BG on page 63, though BG uses the language of arcs, which we will investigate later in this lab.
SPECIAL CASE: If AB is a diameter of circle c, and ACB is an inscribed angle, what does the Inscribed Angle Theorem say in this case. Have you seen this before?
How are the angles related? When are the angles equal? How does this follow from the inscribed angle theorem?
When are the angles not equal? When the angles are not equal, how are they related? How does this follow from the inscribed angle theorem?
In a new figure, draw a circle c and three points inside or on the circle c. Draw the angle CAB, consisting of rays AC and AB. The points of the circle inside the angle form an arc. The is called the arc of c subtended by the angle CAB.
Draw this figure of the angle CAB inside the circle. Let P and Q be the intersections of the circles with the rays. Construct the arc on the circle. This arc is the arc subtended by angle CAB.
But now drag point B so that B moves to the other side of AC. Then the arc constructed becomes the arc or points OUTSIDE the angle instead of inside.
There is another way to construct an arc in Sketchpad. In a figure, select 3 points and choose Construct > Arc Through 3 Points.
Apply this construction to the subtended angle problem. Bisect angle CAB and let the intersection of the bisector with the circle be T. Then construct the arc through P, T, Q (chosen in that order). This will always be the arc subtended by the angle. Try it and see.
Hide the points P and Q and also the angle bisector and point T. Make a Tool called Subtended Arc that takes as inputs the points C, A, B and the circle.
Application 1
In this figure, mark the equal angles and find some similar triangles. Write some equal ratios or products resulting from the similar triangles.
Application 2
If a quadrilateral ABCD is inscribed in a circle, the angles BAD and DCB are both inscribed angles with the same end points. What is the relationship between these angles? What is the relationship between the angles at vertices B and D?
Use your Arc Subtended tool to show the arcs subtended by angles BAD and DCB. Do these arcs explain the relationship?
Application 3: Secants
Draw a circle and a point P outside the circle. Then draw two lines through E intersecting the circle in A and B for one line and C and D for another line. (Such lines are called secant lines.)
Application 4: Regular polygons
Draw a circle c with center O through R. Construct a point A on the circle.