Unit 2 Circles: A Geometric Perspective
Lesson 1
Learning Focus
Find the center of rotation for a given pair of pre-image/image figures.
Lesson Summary
In this lesson, we examined the question: How do you find the center of a rotation if you are only given the pre-image and image figures? This question led us to a strategy involving perpendicular bisectors of segments, as we drew upon the theorem that points on the perpendicular bisector of a segment are equidistant from the endpoints, and its converse, points equidistant from the endpoints of a segment lie on the perpendicular bisector. We were able to find a connection between perpendicular bisectors and the rotation transformation. This connection allowed us to prove a theorem about chords and the central angles formed by drawing radii to the endpoints of the chords.
Lesson 2
Learning Focus
Prove that all circles are similar.
Lesson Summary
In this lesson, we learned how to demonstrate that two circles are similar. One method was to translate one circle so that it coincides with the other circle. Then we can dilate the smaller circle about this common center until it coincides with the outer circle. A second method involved finding the center of dilation that would carry one circle onto the other. The formulas we use to find circumference or area of circles are dependent upon the fact that all circles are similar.
Lesson 3
Learning Focus
Examine the relationship between inscribed angles and the intercepted arc.
Lesson Summary
In this lesson, we learned about cyclic polygons—polygons whose vertices all lie on the circumference of a circle. All triangles are cyclic, and some quadrilaterals are. We used cyclic polygons to make and prove a conjecture about the measure of an inscribed angle relative to its intercepted arc. Once we understood this relationship, we could use it to make a conjecture about opposite angles in a cyclic quadrilateral.
Lesson 4
Learning Focus
Examine the relationships between circumscribed angles and circles.
Lesson Summary
In this lesson, we learned about a class of quadrilaterals that are called kites. Kites occurred in our work when a circle was inscribed within a triangle. The segments drawn from the vertex of the circumscribed angle to the points of tangency on the circle, along with the two radii of the circle drawn to the points of tangency, form the sides of the kite. By examining features of the kite, we were able to develop a formula for the measure of the circumscribed angle, relative to the intercepted arc.
Lesson 5
Learning Focus
Apply circle geometry theorems in various contexts.
Lesson Summary
In this lesson, we applied theorems about inscribed and circumscribed angles of a circle to find the lengths of many segments associated with the circle, including the lengths of the tangent segments drawn from the vertex of a circumscribed angle to the points of tangency on the circle. To find these lengths, we often had to draw upon right triangle trigonometry.
Lesson 6
Learning Focus
Find formulas for arc length and area of a sector of a circle.
Lesson Summary
In this lesson, we found a relationship between arc length and the area of a sector of a circle bounded by the arc and the two radii of the circle drawn to the endpoints of the arc.
Lesson 7
Learning Focus
Develop a new unit for measuring angles.
Lesson Summary
In this lesson, we learned about a new unit for measuring angles called a radian. We developed this new unit of angle measurement by examining the ratio of arc length to radius for various central angles and for various distances from the center of the circle.
Lesson 8
Learning Focus
Measure angles in radians.
Lesson Summary
In this lesson, we learned how to approximate the size of an angle measured in radians, and we learned the radian measure for some familiar angles measured in degrees, such as 90° and 180°. We also learned how to convert between degree and radian measures.
Lesson 9
Learning Focus
Find the equation of a circle.
Lesson Summary
In this lesson, we derived the equation of a circle. We learned that the equation of a circle describes all the points a given distance from the center. Like the distance formula, it is based on the Pythagorean theorem.
Lesson 10
Learning Focus
Write and graph the equation of a circle.
Find the center and radius of a circle in general form.
Lesson Summary
In this lesson, we learned to write equations of circles in both standard form and general form. We used the process of completing the square to change an equation from standard form to general form.
Lesson 11
Learning Focus
Apply understanding of circles and their equations to new situations.
Lesson Summary
In this lesson, we solved problems about circles that required us to use graphs and formulas such as the Pythagorean theorem, the distance formula, and the midpoint formula. We found it useful to use the equation of the circle to find points on the circle or to determine that a point is not on a circle. Sometimes it was useful to change forms of the equation to find more information about the circle from the equation.