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Unit Narrative
In grade 7, students used formulas for circumference and area of circles to solve problems. Earlier in this course, students used straightedges and compasses for constructions, studied similarity and proportional reasoning, and proved theorems about various angle relationships. This unit builds on all of these skills and concepts to investigate the geometry of circles more closely.
In the rst set of lessons, students rigorously de ne radian measure and apply proportional reasoning to solve problems involving circular arc length and sector area. Students revisit the proof that all circles are similar and use proportional reasoning to observe several ratios in circles that are invariant under dilation. Students connect angles and circles by observing that the ratio of arc length to radius of a central angle doesn't depend on the size of the circle, and the ratio increases as the angle takes up a larger portion of the circle. This gives a new way to measure an angle: draw a circle of any size at the vertex of the angle and measure the ratio of arc length to radius. This ratio is known as the radian measure of the angle. Observing ratios that are invariant under dilation is precisely the same line of reasoning that led to de ning trigonometric ratios in an earlier unit. Students then solve problems involving arc length and sector area to develop uency with radian measure. This is important for the transition towards Algebra 2, where they will work with radian measure to de ne trigonometric functions in terms of the unit circle. However, students still have ample opportunity to maintain their uency with degree measure.
Next, students explore the various relationships between angles and segments in circles, leading to the construction of incircles and circumscribed circles. Students observe that inscribed angles are half the measure of their associated central angles. They develop the concept that the shortest path between a point and a line is the segment connecting them that is perpendicular to given line, and they de ne the distance between a point and a line as the length of that shortest path. This allows them to prove that the radius of a circle and a line tangent to the circle are perpendicular at the point of tangency, and that the bisector of an angle is the set of points equidistant to the rays that de ne it. Students build on these concepts to construct the incenter and incircle of any triangle. Then, students are presented with the question of whether it is possible to construct a
circle outside of any triangle that goes through all of its vertices. They use what they know about perpendicular bisectors from an earlier unit to construct the circumcenter and circumscribed circle of any triangle. This idea is extended to quadrilaterals, where students nd that only certain quadrilaterals have circumscribed circles. They prove that these cyclic quadrilaterals must have supplementary opposite angles.
In the nal lesson, students apply what they have learned about relationships between distances and angles in circles to solve problems in context.
Unit 7 Table of Contents 5