About This Chapter
Standard: Prove that all circles are similar. (CCSS.Math.Content.HSG-C.A.1)
Standard: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (CCSS.Math.Content.HSG-C.A.2)
Standard: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (CCSS.Math.Content.HSG-C.A.3)
Standard: Construct a tangent line from a point outside a given circle to the circle. (CCSS.Math.Content.HSG-C.A.4)
Standard: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (CCSS.Math.Content.HSG-C.B.5)
Standard: Derive the equation of a circle of the given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (CCSS.Math.Content.HSG-GPE.A.1)
About This Chapter
An understanding of circles enables students to apply theorems to find the center and radius and derive equations regarding circles. Students who master these concepts will be able to find arc lengths and areas of sectors and circles. They can to prove various properties of circles, including that all circles are similar.
The lessons in this standard cover these topics:
- Proving the similarity of all circles
- Identifying and describing the relationships among inscribed angles, radii and chords
- Constructing the inscribed and circumscribed circles of a triangle, and proving properties of angles for a quadrilateral inscribed in a circle
- Constructing a tangent line from a point outside a given circle to the circle
- Deriving through the use of similarity the fact that the length of an arc intercepted by an angle is proportional to the radius and defining the radian measure as the constant of proportionality; deriving the formula for the area of a sector
- Deriving the equation of a circle of given center and radius using the Pythagorean Theorem; completing the square to find the center and radius of a circle given by an equation
Students can demonstrate their mastery of this standard by being able to prove the similarity of all circles and derive equations relating to circles. The standard and these lessons can prepare students for college and possible careers in many fields such as engineering, architecture and astronomy.
How to Use These Lessons in Your Classroom
The following are some tips on how you can incorporate these lessons on circles into your curriculum to help meet the common core standards:
Pre-Quiz and Post-Quiz Lessons
Ask the students to take the quiz found with the Finding the Center and Radius of a Circle by Completing the Square lesson. Watch the lesson together as a class and discuss it. Have the students take the quiz again and go over their answers to assess their understanding.
Completing the Square
Have the students view the lesson on How to Complete the Square as a homework assignment. In class, review this topic's practice lesson as a group.
Inscribed Angles, Radii and Chords
Give paper plates to the students and ask them to determine the center point. Have them draw lines depicting radii and chords. Some chords should pass through the center and others should not. Ask them to measure the various lines and note any relationships they observe.
1. How to Complete the Square
Completing the square can help you learn where the maximum or minimum of a parabola is. If you're running a business and trying to make some money, it might be a good idea to know how to do this! Find out what I'm talking about here.
2. Completing the Square Practice Problems
Completing the square is one of the most confusing things you'll be asked to do in an algebra class. Once you get the general idea, it's best to get in there and actually do a few practice problems to make sure you understand the process. Do that here!
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