Circles in Euclidean Geometry, Part I
... is not tangent to circle C( O, r ) at point A. Then, there is a second point C where line l intersects circle C( O, r ) . Then, ∠ ACB is inscribed in a semicircle, and so, by Corollary 4.5.12, ∠ ACB is a right angle. Also, ∠ BAC is a right angle since line l is perpendicular to the diameter AB . Thu ...
... is not tangent to circle C( O, r ) at point A. Then, there is a second point C where line l intersects circle C( O, r ) . Then, ∠ ACB is inscribed in a semicircle, and so, by Corollary 4.5.12, ∠ ACB is a right angle. Also, ∠ BAC is a right angle since line l is perpendicular to the diameter AB . Thu ...
Geometry by Jurgensen, Brown and Jurgensen
... Theorem 5-8: If two lines are parallel, then all points on one line are equidistant from the other line. Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Theorem 5-10: A line that contains the midpoint of o ...
... Theorem 5-8: If two lines are parallel, then all points on one line are equidistant from the other line. Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Theorem 5-10: A line that contains the midpoint of o ...
KUD Organizer
... Geometry is the investigation of the relationships and patterns that exist within shapes. Two-dimensional geometric figures are representations of our three-dimensional world. Experimenting with and investigating the relationships between two- and three-dimensional geometric figures connects and int ...
... Geometry is the investigation of the relationships and patterns that exist within shapes. Two-dimensional geometric figures are representations of our three-dimensional world. Experimenting with and investigating the relationships between two- and three-dimensional geometric figures connects and int ...
Chapter 3 Notes - cloudfront.net
... 3.7 Perpendicular Lines in the Coordinate Planes * Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Note: vertical and horizontal lines are perpendicular to each other ...
... 3.7 Perpendicular Lines in the Coordinate Planes * Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Note: vertical and horizontal lines are perpendicular to each other ...
Chapter 3 Notes - cloudfront.net
... 3.7 Perpendicular Lines in the Coordinate Planes * Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Note: vertical and horizontal lines are perpendicular to each other ...
... 3.7 Perpendicular Lines in the Coordinate Planes * Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Note: vertical and horizontal lines are perpendicular to each other ...
Analyzing Isosceles Triangles
... A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60 degrees. ...
... A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60 degrees. ...
Applications of Gauss Law
... hollow cylinder, several coaxial cylindrical shells (like a coaxial cable), as well as more general systems which are uniform and infinitely long in one dimension and axially symmetric in the other two dimensions. In cylindrical coordinates (rc , φ, z), dQ = ρ(rc ) × dV = ρ(rc ) × drc × rc dφ × dz ...
... hollow cylinder, several coaxial cylindrical shells (like a coaxial cable), as well as more general systems which are uniform and infinitely long in one dimension and axially symmetric in the other two dimensions. In cylindrical coordinates (rc , φ, z), dQ = ρ(rc ) × dV = ρ(rc ) × drc × rc dφ × dz ...
Momentum Problems Set1(12) Solutions
... Since momentum is conserved during the interaction between the car and the Earth, and initially the total momentum of the Earth-car system was 0 N∙s, the final momentum must also be 0 N∙s. The only way to achieve this is for the Earth to have the same magnitude of momentum as the car (but in the opp ...
... Since momentum is conserved during the interaction between the car and the Earth, and initially the total momentum of the Earth-car system was 0 N∙s, the final momentum must also be 0 N∙s. The only way to achieve this is for the Earth to have the same magnitude of momentum as the car (but in the opp ...
Noether's theorem
Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.