Chapter 1 - South Henry School Corporation
... 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas gi ...
... 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas gi ...
Geometry
... axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for d ...
... axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for d ...
10.2 Arcs and Chords
... Theorem 10.7 • In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. • AB CD if and only ...
... Theorem 10.7 • In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. • AB CD if and only ...
Bloomfield Prioritized Standards Grades 9
... Make geometric constructions: CC.9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle ...
... Make geometric constructions: CC.9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle ...
Geometry - Caverna Independent Schools
... 1. Verify experimentally the properties of dilations given by a center and a scale factor: 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as ...
... 1. Verify experimentally the properties of dilations given by a center and a scale factor: 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as ...
An Elementary Introduction to the Hopf Fibration
... can one see any connection with physical rotations, as we have claimed? The work in Hopf’s paper [8] was an early achievement in the modern subject of homotopy theory. In loose terms, homotopy seeks to understand those properties of a space that are not altered by continuous deformations. One way to ...
... can one see any connection with physical rotations, as we have claimed? The work in Hopf’s paper [8] was an early achievement in the modern subject of homotopy theory. In loose terms, homotopy seeks to understand those properties of a space that are not altered by continuous deformations. One way to ...
Lesson 1: Thales` Theorem
... RADIUS: May refer either to the line segment joining the center of a circle with any point on that circle (a radius) or to the length of this line segment (the radius). DIAMETER: May refer either to the segment that passes through the center of a circle whose endpoints lie on the circle (a diameter) ...
... RADIUS: May refer either to the line segment joining the center of a circle with any point on that circle (a radius) or to the length of this line segment (the radius). DIAMETER: May refer either to the segment that passes through the center of a circle whose endpoints lie on the circle (a diameter) ...
Lesson 13: The Inscribed Angle Alternate—A Tangent Angle
... We have shown that the inscribed angle theorem can be extended to the case when one of the angle’s rays is a tangent segment and the vertex is the point of tangency. The Example develops another theorem in the inscribed angle theorem’s family: the angle formed by the intersection of the tangent line ...
... We have shown that the inscribed angle theorem can be extended to the case when one of the angle’s rays is a tangent segment and the vertex is the point of tangency. The Example develops another theorem in the inscribed angle theorem’s family: the angle formed by the intersection of the tangent line ...
Problem of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, ""Tangencies""); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2) and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8 = 23.In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN.Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method.