Segments in Circles
Segments in circles from tangent and secant lines.
Table of Contents
- Tangent Lines
- Tangent Lines
- Practice
- Secant Lines
- Secant Lines - Intersect in the Circle
- Secant Lines - Intersect Outside of the Circle
Tangent Lines
What is a Tangent Line?
A line that is tangent to the circle is a line that touches (intersects) the circle at exactly one point. That's it - no more, no less, only one. [br][br]A tangent to Circle A has been created in the diagram below. Move the the [b]point of tangency[/b] (the point where the line touches the circle) around the circle and pay close attention to the measure of [math]\angle ABC[/math] as you move it around. This is your first important property of tangent lines.
Tangent Property 1
Using your best and most specific vocabulary, how would you describe the property that is demonstrated in the diagram above?[br][br][b]Note for all typed-in answers[/b]: If your answer shows as incorrect from the program, make sure to compare it to the official answer. Your answer may be very close and a good answer, but the computer will only mark it as "correct" if it is exactly the same.
Two Tangents
Use the diagram below to investigate what happens when two tangents from the same circle intersect. [br][br]In this diagram, you can move the intersection point (D) and point B. Pay attention to the segments whose measures are showing as you change the circle and/or the tangents.
Tangent Property 2
Summarize what you notice about what happens when two tangent lines from the same circle intersect.
Secant Lines - Intersect in the Circle
What is a Secant Line?
A [b]secant line[/b] is a line that goes through the circle. This means that it intersects the circle in [b]two places[/b], kind of like an entrance and an exit.[br][br]The diagram below shows a circle with two secant lines. Take some time to move points B, C, D, and E around and watch the angle measures. [math]\alpha[/math] is the measure of arc CD, and [math]\beta[/math] is the measure of arc BE. [br][br]On the image below, you can press the play button to watch the construction happen. This is the order of steps:[br][br]1) Plot A - the center of the circle.[br]2) Plot B - a point on the circle.[br]3) Draw the circle through B with A as the center.[br]4) Plot C on the circle.[br]5) Connect B and C to make secant line BC.[br]6) Plot D on the circle.[br]7) Plot E on the circle.[br]8) Connect D and E to make secant line DE.[br]9). Measure arc CD [br]10). Measure arc BE[br]11). Plot F at the intersection of the secants.[br]12). Measure [math]\angle DFC[/math], the angle formed by the intersection of the secants.[br]
Constructing Two Secants
Move the points...
Take some time to move the points around that are on the circle. After you have moved them a little, set them with the following measurements and write down what the measure of angle DFC is in each situation.[br][br]Set the diagram so that arc CD = [math]96^{\circ}[/math] and arc BE = [math]70^\circ[/math]. What is the measure of [math]\angle DFC[/math]?[br][br][br]
Set the diagrams so that arc CD is 40 degrees and arc BE is 100 degrees. [br][br]1. What is the sum of the two arcs?[br][br]2. What is the measure of angle DFC now?
Set the diagram so that arc CD is 28 degrees and arc BE is 52 degrees.[br][br]1. What is the sum of the two arcs?[br][br]2. What is the measure of angle DFC now?
Last one... set the diagram so that arc CD is 110 degrees and arc BE is 70 degrees.[br][br]1. What is the sum of the two arcs?[br][br]2. What is the measure of angle DFC now?
Pattern?
Did you notice a pattern in the angles that you found? [br][br]If you can describe it in an equation, try to do that!