MAFS.912.G-C.1.1: Prove that all circles are similar.
MAFS.912.G-C.2.5 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.
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.)
|Learning Targets and Learning Criteria|
- prove that all circles are similar by showing that for a dilation centered at the center of a circle, the pre-image and the image have equal central angle measures.
- define similarity as rigid motions with dilations, which preserves angle measures and makes lengths proportional.
- convert degrees to radians using the constant of proportionality.
- recognize radian as an alternative method to define the measure of an angle based on the arc it cuts off.
- identify inscribed angles, circumscribed angles, chords, tangents and secants.
- describe the relationship between inscribed angle or circumscribed angle.
- recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle and that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- use Pythagorean Theorem to prove lines tangent to a circle at its radius.
- Watch the following short video to understand why all circles are similar:
- Watch the following short video which explains what a radian is:
- Watch the following short video which shows how to convert between degrees and radians
- Do IXL U. New! Convert between radians and degrees (this is right after U.3 in the list)
- Go to www.geogebra.com and then in the search bar at the top, type in Inscribed Angle Theorem. A bunch of “tiles” will appear and look for the one that says Inscribed Angle Theorem (VI) by Tim Brzezinski. Experiment with the applet and use the slide tool to learn about the relationship between inscribed angles and central angles. Answer the questions and then watch the silent demo at the bottom of the page.
- Then do IXL U.9 Inscribed Angles
- Go to www.geogebra.com again and in the search bar at the top, type in Inscribed Angle Theorem (Corollary 1) and then you may have to scroll down a few rows of the tiles, but find the one that says Inscribed Angle Theorem (Corollary 1) (Proof without words). Make sure you find the one that says exactly what is in bold (also by Tim Brzezinski) or you won’t be using the correct one. This whole name will not show and there is a corollary 2, so hover over the tile to make sure it says corollary 1. Follow the instructions and play around with this applet to learn about inscribed angles that intercept a diameter of a circle.
- Then do IXL U.10. You will need to recall some things you already know, such as the fact that if a triangle is an isosceles triangle, then the angles across from the two congruent sides are also congruent. You may need to set up an equation using algebraic expressions for some of them!
- Once again, go to www.geogebra.com and this time type in Inscribed Angle Theorem: Corollary 1 (again by Tim Brzezinski). You will see a circle with a pale pink background and three darker pink angles). Experiment with this applet, move the blue dots to change the size of the arc and move the pink dots around so that you understand the relationship between inscribed angles that intercept the same arc.
- Then go into Gradebook and find the assignment called Inscribed Angle Notes and open the attachment and copy these notes into your ISN (this will be page 56). Make sure you include the diagrams (I would find a round object to trace circles!).
- Yes, you are going to go to www.geogebra.com one more time! This time, type in Cyclic Quadrilaterals. Look for the tile with that title that shows it is by Tim Brzezinski. Experiment with that applet to understand the relationship between the opposite angles of a quadrilateral inscribed in a circle.
- Do IXL U.11 on Angles in inscribed Quadrilaterals I. For some of the questions, you will need to think about creating equations to show that two angles are supplementary.
- IXL’s: U. New! Converting between radians and degrees, U.9, U.10, and U.11
- Notes in your ISN on Inscribed Angles