High School Math 3 Unit 5: Circles
... Students continue to expand their understanding of geometry by exploring geometric relationships pertaining to circles. As was the case in Math 1 and earlier in Math 2, attributes of circles observed at earlier grades will now be looked at more precisely through proof. Many of the geometric relation ...
... Students continue to expand their understanding of geometry by exploring geometric relationships pertaining to circles. As was the case in Math 1 and earlier in Math 2, attributes of circles observed at earlier grades will now be looked at more precisely through proof. Many of the geometric relation ...
Dividing a decimal by a whole number
... Topic VII Properties of Circles Vocabulary Circle - Is the set of all points in a plane equidistant from a given point called the center of the circle. Radius: Any segment with endpoints that are the center of the circle and a point on the circle. Chord: Segments with endpoints that are on the circ ...
... Topic VII Properties of Circles Vocabulary Circle - Is the set of all points in a plane equidistant from a given point called the center of the circle. Radius: Any segment with endpoints that are the center of the circle and a point on the circle. Chord: Segments with endpoints that are on the circ ...
Syllabus for Accelerated Geometry
... analyze geometric figures using deductive reasoning, make conjectures and formulate hypotheses, draw conclusions and make connections with other mathematical concepts, and model situations geometrically as a problem solving strategy. Algebraic and geometric skills are integrated throughout the curri ...
... analyze geometric figures using deductive reasoning, make conjectures and formulate hypotheses, draw conclusions and make connections with other mathematical concepts, and model situations geometrically as a problem solving strategy. Algebraic and geometric skills are integrated throughout the curri ...
Circle
... ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1.____ (10-1) Circles and Circumference – Day 1- Pages 526-527 16-20, 32-54 even 2. ____ (10-2) Angles and Arcs – Day 1- Pages 533-535 14 – 31, 32 -42 even, 58 3. ____ (10-2) Angles and Arcs ...
... ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1.____ (10-1) Circles and Circumference – Day 1- Pages 526-527 16-20, 32-54 even 2. ____ (10-2) Angles and Arcs – Day 1- Pages 533-535 14 – 31, 32 -42 even, 58 3. ____ (10-2) Angles and Arcs ...
Construction 12: Construct a circle circumscribed about a triangle. 1
... 2. Bisect one of the radii. Point X is the midpoint. 3. Measure XA (midpoint of radius to the endpoint of another radius). Mark off the same length on the original diameter from the midpoint of the radius. Call this point Y. 4. Measure the length of AY. Using this length, mark off consecutive arcs o ...
... 2. Bisect one of the radii. Point X is the midpoint. 3. Measure XA (midpoint of radius to the endpoint of another radius). Mark off the same length on the original diameter from the midpoint of the radius. Call this point Y. 4. Measure the length of AY. Using this length, mark off consecutive arcs o ...
r= radius - WorkNotes
... The point where two or more rays meet; the point of intersection of two sides of a polygon; the point of intersection of three or more edges of a solid figure; the top point of a cone. ...
... The point where two or more rays meet; the point of intersection of two sides of a polygon; the point of intersection of three or more edges of a solid figure; the top point of a cone. ...
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.