The steps in the construction result in a line m through the given point A that is parallel to the given line n. Which statement justifies why the constructed line is parallel to the given line?
Best Results From Wikipedia Yahoo Answers Youtube From Wikipedia Polar coordinate system In mathematicsthe polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
The fixed point analogous to the origin of a Cartesian system is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.
History The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologer Hipparchus BCE created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.
In On SpiralsArchimedes describes the Archimedean spirala function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately.
The calculation is essentially the conversion of the equatorial polar coordinates of Mecca i. Around CE, he was the first to describe a polar equi- azimuthal equidistant projection of the celestial sphere. There are various accounts of the introduction of polar coordinates as part of a formal coordinate system.
Saint-Vincent wrote about them privately in and published his work inwhile Cavalieri published his in with a corrected version appearing in Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral.
Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs. In Method of Fluxions writtenpublishedSir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.
In the journal Acta EruditorumJacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.
The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.
They have the advantage that the coordinates of a point, even those at infinity, can be represented using finite coordinates.
Often formulas involving homogeneous coordinates are simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer visionwhere they allow affine transformation s and, in general, projective transformation s to be easily represented by a matrix.
If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general, one more than the dimension of the projective space being considered.
For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point on the projective plane. Introduction The projective plane can be thought of as the Euclidean plane with additional points, so called points at infinity, added.
There is a point at infinity for each direction, informally defined as the limit of a point that moves in that direction away from a fixed point. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. A given point on the Euclidean plane is identified with two ratiosso the point corresponds to the triple where.
Such a triple is a set of homogeneous coordinates for the point. The equation of a line through the point may be written where l and m are not both 0.
In parametric form this can be written. In homogeneous coordinates this becomes. In the limit as t approaches infinity, in other words as the point moves away fromZ becomes 0 and the homogeneous coordinates of the point become.
So are defined as homogeneous coordinates of the point at infinity corresponding to the direction of the line. Any point in the projective plane is represented by a triplecalled the homogeneous coordinates of the point, where X, Y and Z are not all 0.
The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor. Conversely, two sets of homogeneous coordinates represent the same point only if one is obtained from the other by multiplying by a common factor.
When Z is not 0 the point represented is the point in the Euclidean plane. When Z is 0 the point represented is a point at infinity. Note that the triple is omitted and does not represent any point.
The origin is represented by. Notation Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates.
The use of colons instead of commas, for example x:GEOMETRY – REVIEW FOR MIDTERM The midterm exam for period 4 is on 1/28, AM to AM. The exam will 6 false plane ABC 6 plane FDE true and are skew lines. true and are skew lines. false Lessons Write an equation in point-slope form of the line that contains the given points.
21– See margin. A(4, 2). Analytic Geometry With Introduction to Vector Analysis - Free download as PDF File .pdf) or read online for free. Answer: x–2y–6=0 Slope-intercept form: y = mx + c is the equation of a straight line whose slope is m & which makes an intercept c on the y axis.
x 2 y 2 1 = 0 x3 y3 1 (iii) AC = AB + BC or AB ~ BC (iv) A divides the line segment BC in some ratio.
how a line can be represented with the help of an algebraic equation. Download as PDF, TXT or read online from Scribd. Flag for inappropriate content. Descarga. Point slop is one of the method used to find the straight line equation. Use the below point slope form calculator to calculate the equation of the straight line by .
Geometry (1) - Free download as Word Doc .doc /.docx), PDF File .pdf), Text File .txt) or read online for free. OR if we want to write the equation in the slope-intercept form: From two given points on the line. From the equation of the line in slope-intercept form From the equation of the line in point-slope form The equation of a.