In geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by, two lines In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some or planes In mathematics, a plane is a flat surface Chyea. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry (or a line and a plane), are considered perpendicular (or orthogonal In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός , meaning "straight", and γωνία (gonia), meaning "angle") to each other if they form congruent In geometry, two of sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions adjacent angles In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (a T-shape). The term may be used as a noun A noun can co-occur with an article or an attributive adjective. Verbs and adjectives can't. In the following, an asterisk in front of an example means that this example is ungrammatical or adjective In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun, giving more information about the noun or pronoun's referent. Collectively, adjectives form one of the traditional English eight parts of speech, though linguists today distinguish adjectives from words such as determiners that also used to be considered. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B. Note that by definition, a line In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some is infinitely long, and strictly speaking AB and CD in this example represent line segments In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge if they are adjacent vertices, or of two infinitely long lines. Hence the line segment AB does not have to intersect line segment CD to be considered perpendicular lines, because if the line segments are extended out to infinity, they would still form congruent adjacent angles.
If a line is bending to another as in Figure 1, all of the angles created by their intersection are called right angles (right angles measure ½π π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which radians The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit. The SI, or 90° A degree , usually denoted by ° (the degree symbol), is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere, such as Earth (see Geographic coordinate system), Mars, or the). Conversely, any lines that meet to form right angles are perpendicular.
In a coordinate plane, perpendicular lines have opposite reciprocal slopes. A horizontal line has slope equal to zero while the slope of a vertical line is described as undefined or sometimes ±infinity. Two lines that are perpendicular would be denoted as ABCD
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Numerical criteria
In terms of slopes
In a Cartesian coordinate system A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length, two straight lines L and M may be described by equations.
- L : y = ax + b
- M : y = cx + d
as long as neither is vertical. Then a and c are the slopes The Grade or Slope of a physical feature, topographic landforms or constructed elements, refers to the amount of inclination of that surface where zero indicates gravitational level. A larger number indicates higher or steeper degree of "tilt" or grade. Often slope is calculated as a ratio of "rise over run" in which run is the of the two lines. The lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1.
Construction of the perpendicular
Fig. 2: Construction of the perpendicular (blue) to the line AB through the point P.To make the perpendicular to the line AB through the point P using compass and straightedge Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass, proceed as follows (see Figure 2).
- Step 1 (red): construct a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius with center at P to create points A' and B' on the line AB, which are equidistant Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific from P.
- Step 2 (green): construct circles centered at A' and B', both passing through P. Let Q be the other point of intersection of these two circles.
- Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.
To prove that the PQ is perpendicular to AB, use the SSS congruence theorem In geometry, two of sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem In geometry, two of sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions for triangles OPA' and OPB' to conclude that angles POA and POB are equal.
In relationship to parallel lines
Fig. 3: Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c.As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that:. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.
In Figure 3, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:
- One of the angles in the diagram is a right angle.
- One of the orange-shaded angles is congruent to one of the green-shaded angles.
- Line 'c' is perpendicular to line 'a'.
- Line 'c' is perpendicular to line 'b'.
Finding the perpendiculars of a function
Algebra
In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a⁄b is b⁄a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one of the original slope. It is helpful to memorize the slogan "to find the slope of a perpendicular line, flip the fraction and swap the sign." Recall that any whole number a is itself over one, and can be written as (a/1)
To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.
Calculus
First find the volume of the function. This will be the slope (m) of any curve at a particular point (x, y). Then, as above, solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.
Perpendicular symbol
The perpendicular symbol is . For example, indicates that line AB is perpendicular to line CD.
In the Unicode Unicode is a computing industry standard for the consistent representation and handling of text expressed in most of the world's writing systems. Developed in conjunction with the Universal Character Set standard and published in book form as The Unicode Standard, the latest version of Unicode consists of a repertoire of more than 107,000 character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range. It often looks the same as the "up tack" symbol (U+22A5), but is a different traits.
See also
- Orthogonality In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός , meaning "straight", and γωνία (gonia), meaning "angle"
- Perpendicular component (of a vector)
- Surface normal A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force,
- Parallel (geometry) Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or meet are called parallel lines
External links
- Definition: perpendicular With interactive animation
- How to draw a perpendicular bisector of a line with compass and straight edge Animated demonstration
- How to draw a perpendicular at the endpoint of a ray with compass and straight edge Animated demonstration
Categories: Geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, | Orientation
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