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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 deRne 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 deRne 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 Rnd that only certain quadrilaterals have circumscribed circles. They prove that these cyclic quadrilaterals must have supplementary opposite angles.
In the Rnal lesson, students apply what they have learned about relationships between distances and angles in circles to solve problems in context.
S3 Conditional Probability
In grade 7, students had the opportunity to learn about chance processes and examine probability models using simulation as well as analysis of sample spaces. In this unit, students extend that knowledge by considering events that are combined in various ways including both occurring, at least one occurring, and one event happening under the condition that the other happens as well.
The unit begins with leveraging diUerent models for understanding sample spaces and probability using tables, trees, lists, and Venn diagrams. Venn diagrams help students visualize various subsets of the sample space such as “A and B,” “A or B,” “not A,” and other combinations of events. Tables are used to show the probability of various combinations of categories occurring as well as to introduce the relationship between probabilities of some combinations of events using the Addition Rule,
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Conditional probability is discussed and explored using several games and connections to everyday situations. In particular, the Multiplication Rule is used to determine conditional probabilities as well as independence of events A and B. Independence is further explored using everyday language as well as through the equation when events A and B are independent.
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Course Guide