Geometry Unit 9 Plan (July 2015)
... Use coordinate to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). ...
... Use coordinate to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). ...
Chapter 18/19 Study Guide - East Penn School District
... obtuse angle – an angle that measures greater than 90˚ and less than 180˚ straight angle – an angle that measures 180˚ parallel lines – are lines in a plane that are always the same distance apart. intersecting lines – are lines a plane that cross at exactly one point. perpendicular lines – are two ...
... obtuse angle – an angle that measures greater than 90˚ and less than 180˚ straight angle – an angle that measures 180˚ parallel lines – are lines in a plane that are always the same distance apart. intersecting lines – are lines a plane that cross at exactly one point. perpendicular lines – are two ...
Chapter 6 Notes: Circles
... A polygon is circumscribed about a circle if all sides of the polygon are segments tangent to the circle; also the circle is said to be inscribed in the polygon. ...
... A polygon is circumscribed about a circle if all sides of the polygon are segments tangent to the circle; also the circle is said to be inscribed in the polygon. ...
document
... chords subtend (cut off) equal arcs (arcs of equal length) In the same circle or in congruent circles, equal central angles subtend (cut off) equal arcs In the same circle or congruent circles, two central angles have the same ratio as the arcs that are subtended (cut off) by the ...
... chords subtend (cut off) equal arcs (arcs of equal length) In the same circle or in congruent circles, equal central angles subtend (cut off) equal arcs In the same circle or congruent circles, two central angles have the same ratio as the arcs that are subtended (cut off) by the ...
Geometry Scope and Sequence
... Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of cong ...
... Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of cong ...
Study Guide
... - Use right triangle trigonometry to solve application problems: p. 536 ex 5, p540 #67 ( answer: A) - Identify and use angles of elevation and depression with right triangle trig: p. 545 ex 2, 3; p 547 #9 - Law of Sines and Law of Cosines: p 551-4. EXAMPLES: 2AB and 3AB o Remember, if it’s a right t ...
... - Use right triangle trigonometry to solve application problems: p. 536 ex 5, p540 #67 ( answer: A) - Identify and use angles of elevation and depression with right triangle trig: p. 545 ex 2, 3; p 547 #9 - Law of Sines and Law of Cosines: p 551-4. EXAMPLES: 2AB and 3AB o Remember, if it’s a right t ...
Angles and Circles
... In classical constructions, you are allowed to use only compasses and a straight edge, i.e., one side of a ruler whose markings you must ignore. Typically you are given certain data to work with. For example, lengths are normally given by a line segment (which you can copy using the compasses). 4. S ...
... In classical constructions, you are allowed to use only compasses and a straight edge, i.e., one side of a ruler whose markings you must ignore. Typically you are given certain data to work with. For example, lengths are normally given by a line segment (which you can copy using the compasses). 4. S ...
Unit 4 Circles Geometry ACC - Long Beach Unified School District
... • Students will identify and explain why inscribed angles on a diameter are right angles. • Students will identify and explain why the radius of a circle is perpendicular to the tangent where the radius intersects the circle. • Students will prove properties of angles for a quadrilateral inscribed i ...
... • Students will identify and explain why inscribed angles on a diameter are right angles. • Students will identify and explain why the radius of a circle is perpendicular to the tangent where the radius intersects the circle. • Students will prove properties of angles for a quadrilateral inscribed i ...
Mathematics Chapter: Tangents to Circles - JSUNIL tutorial
... 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches all the three semicircles. Solution: Let the required radius be r cm. O1O3 = Radius of smaller semicircle + r = 24/4 + r = 6 + r O1C = Radius of smalle ...
... 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches all the three semicircles. Solution: Let the required radius be r cm. O1O3 = Radius of smaller semicircle + r = 24/4 + r = 6 + r O1C = Radius of smalle ...
Postulates - Geneseo Migrant Center
... This is similar to Postulate 8, in that there can only be one middle to something. The same is true for angles. For any angle, there is only one way to cut it in half. ...
... This is similar to Postulate 8, in that there can only be one middle to something. The same is true for angles. For any angle, there is only one way to cut it in half. ...
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.