Unit 5 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 circle. 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.