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Circles With and Without Coordinates

Circles With and Without Coordinates

In this unit, students will interact with concentric circles to discover that they are similar through dilation. Students will find scale factors that correspond to enlargements and to reductions. They will also interact with non-concentric circles to realize that non-concentric circles are also similar through translation and dilation. They will find the transformation or sequence of transformations that maps one circle onto another. In conclusion, it will be shown that all circles are similar through similarity transformations.

Inscribed and central angles of a circle will be defined, as well as their intercepted arc. Students will interact with inscribed angles to find that the measure of an inscribed angle whose intercepted arc is a semicircle is 90 degrees. Students will also interact with inscribed and central angles, learning that if they intercept the same arc, then the measure of the central angle is twice the measure of the inscribed angle. Moreover, the tangent line to a circle will be defined and students will be able to use interactive graphs to realize that a tangent line to a circle is always perpendicular to the radius at the point of tangency. Furthermore, circumscribed angles will be defined and students will have the opportunity to explore them through interactive diagrams. Several theorems will be stated, proved, and used to find missing measures.

  • The Inscribed Angle Theorem
  • The Inscribed Angles of a Circle Theorem
  • The Tangent to a Circle Theorem
  • The Circumscribed Angle Theorem

Inscribed and circumscribed circles of a triangle will be defined and students will have the opportunity of investigating them through interactions. The incenter, circumcenter, and centroid of a triangle will also be defined, including a recollection of the concepts of angle bisector, perpendicular bisector, and median. Their properties will be used to solve problems. Students will learn the method for drawing both the inscribed and circumscribed circles of a triangle, and will use them to solve real-life problems.

Students will investigate and interact with inscribed quadrilaterals and realize that opposite angles are supplementary. Inscribed quadrilaterals will be defined along with a mention of the alternative name, cyclic quadrilateral. Students will see the formal proof that opposite angles in an inscribed quadrilateral are supplementary, and use this fact to find missing angle measures. The Cyclic Quadrilateral Exterior Angle Theorem will be stated, proved, and used to find missing angle measures. Students will also learn that the perpendicular bisectors of a cyclic quadrilateral are concurrent.

Recalling the definition of a tangent line to a circle, students will learn how to draw the tangent through an outer point, and use this construction to solve problems. The relationship between tangents and lines of symmetry will be investigated. The External Tangent Congruence Theorem will be stated, proved, and used. Applications of tangent lines to circles in real life will be seen.

This unit will also explore the relationship between radian measure and arc length. Students will learn about radians and how to convert from radians to degrees and vice versa. They will calculate radian measures and compare radian measures of concentric circles. Moreover, radians will be used to calculate arc lengths and angle measures.

In addition to their work with arc lengths, students will also work with the area of circle sectors. The definition of a circle sector will be stated and the formula for the area of a sector will be proved. Students will use this formula to solve sector problems.

In the Cartesian coordinate system, students will use the Distance Formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, students will identify the radius and the center, and will draw the graph in the coordinate plane. Students will algebraically verify whether a point is on a circle, and write the standard equation of a circle by completing squares.

Coordinates will be used to write proofs. Properties of parallelograms, circles, and triangles will be proved by using coordinates. Moreover, real life problems involving these geometric shapes will be solved by using coordinates.

Finally, the unit will cover the modeling of real life objects with circles. Properties of circles, radius, and diameter will be used to calculate area and circumference of ordinary objects.