CIR - Circles Module Overview
Introduction to Circles
Take one minute and name as many circular objects as you can think of. How many did you come up with? Most people can name many circular objects off the top of their head, but few can list the many properties and special parts of a circle. With circles appearing so often in everyday life, it is important and helpful for us to understand their special properties. It is these same properties that allow us to solve many every day problems. What is the best location for a cell tower that is supposed to service three different cities? What are the different types of intersections the sun and the horizon may have? What is the circumference of a broken plate that we only have a few pieces of? These are just a few questions that can be answered using circles and their properties.
Essential Questions
- How do we know all circles are similar?
- How can we apply circle properties to real life situations in order to solve problems?
- What are angles, radii, chords, tangents, and secants and how are they related?
- What is the difference between central, inscribed, and circumscribed angles?
- How can we show that an inscribed angle on a diameter is a right angle?
- How do we know that a radius of a circle is perpendicular to a tangent at the point where they both intersect the circle?
- What is the difference between an inscribed and circumscribed circle of a triangle, and how can we construct each?
- How can we prove the various properties of angles of a quadrilateral inscribed in a circle?
- What is meant by the radian measure of an angle?
- What is the connection between the radian measure of an angle and the length of the arc on the unit circle the angle intercepts?
Key Terms
Arc - a portion of the circumference of a circle
Arc measure - the measure of the angle at the center of the circle that creates an arc
Central angle - an angle whose vertex is the center of the circle
Chord - a segment whose endpoints both lie on the circle
Circumcenter - the point of intersection of the perpendicular bisectors of a triangle that is inscribed in a circle; this is also the center of the circle circumscribed around a given triangle
Circumscribed angle - an angle whose vertex is outside the circle and whose sides are made by two intersecting tangent lines to a circle
Circumscribed circle - a circle that passes through all the vertices of a polygon
Common tangent - a line that is tangent to two coplanar circles
Coterminal Angles - share the same initial side and the same terminal side of angles of rotation.
Cyclic quadrilaterals - a quadrilateral inscribed in a circle (all 4 vertices lie on the circle)
Initial Side - the "beginning" side of an angle of rotation, usually on the positive x-axis
Inscribed angle - an angle whose vertex is on the circle and whose sides are made up by chords of a circle
Inscribed circle - a circle contained in a polygon where each side of the polygon is tangent to the circle
Inscribed polygon - a polygon contained inside a circle whose vertices all lie on the circle
Major arc - an arc whose measure is greater than 180 degrees
Minor arc - an arc whose measure is less than 180 degrees
Negative Angle - an angle in standard position is negative when the location of the terminal side results from a clockwise rotation
Point of tangency - the point where a tangent line intersects the circle
Positive Angle - an angle in standard position is positive when the location of the terminal side results from a counterclockwise rotation
Radian - the measure of the central angle of a circle subtended by an arc of equal length to the radius
Reference Angle - the measure of the acute angle formed by the terminal side and the x-axis
Secant line - a line that intersects a circle at exactly two points
Semicircle - a half circle; created by an arc whose measure is equal to 180 degrees
Standard Position - an angle is in standard position when the vertex is at the origin and the initial side lies on the positive x-axis
Tangent line - a line in the plane of the circle that intersects it at exactly one point
Terminal Side - the "ending" side of an angle of rotation
Unit Circle - a circle with a radius of 1 and center at the origin