Unit 8: Circles
In this unit, we will explore circles and arcs, which are slices of the edge of a circle. We use arcs to describe anything that looks like an incomplete circle, such as the feet of a rocking chair or the crust of a slice of pizza. We will learn to classify and measure arcs in this unit.
Completing this unit should take you approximately 3 hours.
Upon successful completion of this unit, the student will be able to:
- identify the various parts of a circle;
- determine arc lengths of a circle; and
- determine the measure of angles created by lines intersecting a circle.
8.1: Language and Notation of a Circle
Before we can begin studying calculations involving circles, we need to understand the definitions and notation we use to describe circles and circular objects.
Watch this video to become familiar with the terms we will use in this unit.
8.2: Tangent Lines
We can draw special lines called tangent lines on circles. A tangent line touches the circle at only one point, and is perpendicular to the radius of the circle drawn to the point where the tangent line touches the circle.
Read this article and watch the videos to learn about the tangent line theorem, which defines tangent lines, and the two tangents theorem, which we use to solve for unknown lengths of line segments. Read examples 1–5 closely to see how to draw tangent lines and how we can use tangent lines to solve for unknown quantities.
Then, complete review questions 1, 2, 4, and 5 and check your answers.
8.3: Arcs of Circles
An arc is a section of a circle.
Watch these videos, which introduce the idea of arcs in circles and give examples of how to determine the angle of a given arc.
8.4: Inscribed Angles
An inscribed angle occurs inside a circle. The angle's vertex is on the circle and the rays of the angle are chords (lines within the circle) of the circle.
Read this brief article, which gives an overview of what inscribed angles are and how to identify them. Watch the video to see how to use your knowledge of triangles to determine the value of inscribed angles.
8.5: Angles of Chords, Secants, and Tangents
Now, we put together what we have learned to calculate more advanced trigonometric ratios. First, we need to study the meaning of chords, or line segments that exist within a circle.
Read this article and watch the video. Pay attention to the examples of how to determine the lengths of chords and their angles. It may help to write down the four chord theorems to help you keep track of them as you work through the problems.
Once you feel comfortable with the definition of chords and their theorems, read this article which details several ways we can calculate the angles formed by chords. Since this page has so much information, read slowly and take time to work through each example. It may be helpful to write down the many theorems to make sure you understand them.
Then, complete review questions 1–10 and check your answers.