Chapter 9
... circular design, and sewed a blue ribbon on the circle that passes through the circle’s centre, O. She wants to sew on two more blue ribbons to the circle, as shown. a) What is the measure of the angle formed at the point on the circle where the two new ribbons meet? b) If the point where the new ri ...
... circular design, and sewed a blue ribbon on the circle that passes through the circle’s centre, O. She wants to sew on two more blue ribbons to the circle, as shown. a) What is the measure of the angle formed at the point on the circle where the two new ribbons meet? b) If the point where the new ri ...
2012-2013 Instructional Curriculum Plan Grade: 10 Course
... http://www.corestandards.org/the-standards/mathematics/high-school-geometry/expressing-geometric-properties-with-equations/ MA.912.G.2.2: Determine the measures of http://www.corestandards.org/the-standards/mathematics/high-school-geometry/congruence/ interior and exterior angles of polygons, Linked ...
... http://www.corestandards.org/the-standards/mathematics/high-school-geometry/expressing-geometric-properties-with-equations/ MA.912.G.2.2: Determine the measures of http://www.corestandards.org/the-standards/mathematics/high-school-geometry/congruence/ interior and exterior angles of polygons, Linked ...
Quadrilaterals Theorems Ptolemy`s Theorem
... 1. If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, the line passes through the incenter 2. If M and N are midpoints of diagonals and incenter = I, then M, I, and N are collinear ...
... 1. If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, the line passes through the incenter 2. If M and N are midpoints of diagonals and incenter = I, then M, I, and N are collinear ...
File
... Compare your results with those of another pair of classmates. What appears to be true about each line segment and its related fold line? What name could you give each fold line? Through which point do both fold lines appear to pass? ...
... Compare your results with those of another pair of classmates. What appears to be true about each line segment and its related fold line? What name could you give each fold line? Through which point do both fold lines appear to pass? ...
11-2 Reteach Arcs and Chords
... 7. a chord one-fourth the length of the circumference 1_ multiplied by the length of the circumference 8. a chord _� Find the length of a chord that intercepts an arc of each given measure. Give your answer in terms of the radius r. Round to the nearest tenth. ...
... 7. a chord one-fourth the length of the circumference 1_ multiplied by the length of the circumference 8. a chord _� Find the length of a chord that intercepts an arc of each given measure. Give your answer in terms of the radius r. Round to the nearest tenth. ...
1Topic
... Euclid was one of the great mathematical thinkers of ancient times. It is known that he was the founder of a school in Alexandria during the reign of Ptolemy I, which lasted from 323 BC to 284 BC. Euclid’s most famous mathematical writing is the Elements. This work is the most complete study of geom ...
... Euclid was one of the great mathematical thinkers of ancient times. It is known that he was the founder of a school in Alexandria during the reign of Ptolemy I, which lasted from 323 BC to 284 BC. Euclid’s most famous mathematical writing is the Elements. This work is the most complete study of geom ...
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.