Geometric Constructions for Love and Money
... 9. Paperfolding constructions use folds in paper. Any constructions of plane Euclidian geometry can be executed by means of creases. 10. Constructions using other instruments like a “tomahawk”, carpenter’s squares, strings, etc. allow for constructions not possible using a ruler and compass. 11. App ...
... 9. Paperfolding constructions use folds in paper. Any constructions of plane Euclidian geometry can be executed by means of creases. 10. Constructions using other instruments like a “tomahawk”, carpenter’s squares, strings, etc. allow for constructions not possible using a ruler and compass. 11. App ...
Algebra 1 GT Lesson Plan
... 25. Given the lengths a and b, construct the following and label it. ...
... 25. Given the lengths a and b, construct the following and label it. ...
List of all Theorems Def. Postulates grouped by topic.
... Theorem 6.3.2: If a line through the center of a circle bisects a chord other than a diameter then it is perpendicular to the chord Theorem 6.3.3: The perpendicular bisector of a chord contains the center of the circle. Theorem 6.3.4: The tangent segments to a circle from an external point are congr ...
... Theorem 6.3.2: If a line through the center of a circle bisects a chord other than a diameter then it is perpendicular to the chord Theorem 6.3.3: The perpendicular bisector of a chord contains the center of the circle. Theorem 6.3.4: The tangent segments to a circle from an external point are congr ...
Circle geometry
... For thousands of years humans have been fascinated by circles. Since they first looked upwards towards the sun and moon, which, from a distance at least, looked circular, humans have created circular monuments to nature. The most famous circular invention, one that has been credited as the most impo ...
... For thousands of years humans have been fascinated by circles. Since they first looked upwards towards the sun and moon, which, from a distance at least, looked circular, humans have created circular monuments to nature. The most famous circular invention, one that has been credited as the most impo ...
Unit 7 – Polygons and Circles Diagonals of a Polygon
... Participants will now investigate the exterior angles of a polygon. It is often difficult for students to understand intuitively that this sum does not depend on the number of sides of the polygon. Participants may use the polygons that they have already constructed to study exterior angles for the ...
... Participants will now investigate the exterior angles of a polygon. It is often difficult for students to understand intuitively that this sum does not depend on the number of sides of the polygon. Participants may use the polygons that they have already constructed to study exterior angles for the ...
Geometric Figures
... As always, the word “line” means a straight line that extends indefinitely in both directions. Consequently, to make sense of the phrase “do not intersect” children must envision extending the segments on their paper indefinitely into space, past distant galaxies — something that is not very concret ...
... As always, the word “line” means a straight line that extends indefinitely in both directions. Consequently, to make sense of the phrase “do not intersect” children must envision extending the segments on their paper indefinitely into space, past distant galaxies — something that is not very concret ...
Q2 - Franklin County Community School Corporation
... G.4.3: Construct triangles congruent to given triangles. G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. G.4.5: Prove and apply theorems involving segments divided proportionally. G.4.6: Prove that triangles are congruent or similar and use the ...
... G.4.3: Construct triangles congruent to given triangles. G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. G.4.5: Prove and apply theorems involving segments divided proportionally. G.4.6: Prove that triangles are congruent or similar and use the ...
Geometry Curriculum - Oneonta City School District
... bisector of a line segment, given the endpoints of the line segment. Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas Solve systems of equations involving one linear equation and one quadratic equ ...
... bisector of a line segment, given the endpoints of the line segment. Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas Solve systems of equations involving one linear equation and one quadratic equ ...
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.