In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
The term reflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.
A figure that does not change upon undergoing a reflection is said to have reflectional symmetry.
Construction
In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
To reflect point P through the line AB using compass and straightedge, proceed .....
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
A translation moves every point of a figure or a space by the same amount in a given direction.
Video: Translation example | Transformations
A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
In Euclidean geometry a transformation is a one-to-one correspondence between two sets of points or a mapping from one plane to another.[1] A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections.
A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
Two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
An example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distance and angles. The unchanged properties are called invariants.More formally, two 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 rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.
This diagram illustrates the geometric principle of angle-angle-side triangle congruence: Given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if angle CAB is congruent with C'A'B' and angle BCA is congruent with B'C'A' and BC is congruent with B'C'
In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects.
Two line segments are congruent if they have the same length.
Two angles are congruent if they have the same measure.
Two circles are congruent if they have the same diameter.
Video: What are Congruent Figures?
In this sense, two plane figures are congruent implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters and areas.
The related concept of similarity applies if the objects have the same shape but do not necessarily have the same size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.)
Determining congruence of polygons
The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
For two polygons to be congruent, they must have an equal number of sides (and hence an equal number—the same number—of vertices). Two polygons with n sides are congruent if and only if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for n sides and n angles.
Video: Congruent Polygons
Congruence of polygons can be established graphically as follows:
First, match and label the corresponding vertices of the two figures.
Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match.
Third, rotate the translated figure about the matched ...................
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles.
Video: External Angle Theorem
In several high school treatments of geometry, the term "exterior angle theorom" has been applied to a different result, namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel postulate will be referred to as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem.
Exterior angles
Video: Exterior Angles of a Polygon
A triangle has three corners, called vertices. The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. In the picture below, the angles ∠ABC, ∠BCA and ∠CAB are the three interior angles of the triangle. An exterior angle is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle ∠ACD is an exterior angle.
Euclid's exterior angle theorem
The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a flawed proof.
Euclid proves the exterior angle theorem by:
construct the midpoint E of segment AC,
draw the ray BE,
construct the point F on ray BE so that E is (also) the midpoint of B and F,
draw the segment FC.
By congruent triangles we can conclude that ∠ BAC = ∠ ECF and ∠ ECF is smaller than ∠ ECD, ∠ ECD = ∠ ACD therefore ∠ BAC is smaller than ∠ ACD and the same can be done for the angle ∠ CBA by bisecting BC.
The flaw lies in the assumption that a point (F, above) lies "inside" angle (∠ ACD). No reason is given for this assertion, but the accompanying diagram makes it look like a true statement. When a complete set of axioms for Euclidean geometry is used (see Foundations of geometry) this assertion of Euclid can be proved.
High school exterior angle theorem
Video: Exterior Angle Theorem (High School EAT) | Two Opposite Interior Angles Add to the Exterior Angle
The high school exterior angle theorem (HSEAT) says that .....
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,
Video: Factorials ; Evaluating Expressions With Factorials
The value of 0! is 1, according to the convention for an empty product.
The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects).
An angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interiorangle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.
If every internal angle of a simple polygon is less than 180°, the polygon is called convex.
In contrast, an exteriorangle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.
Video: What is the Relation of an Exterior Angle of a Triangle with its Interior Angles?
Properties
The sum of the internal angle and the external angle on the same vertex is 180°.
The sum of all the internal angles of a simple polygon is 180(n–2)° where ......
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom.
This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology.
A solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder. If the bases are disks (regions whose boundary is a circle) the cylinder is called a circular cylinder. In some elementary treatments, a cylinder always means a circular cylinder.
The height (or altitude) of a cylinder is the perpendicular distance between its bases.
Right circular cylinders
Video: What is a Right Circular Cone?
The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder.
A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.
Cylindric sections
Cylindric section
A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram. Such a cylindric section of a right cylinder is a rectangle.
Video: Cylindrical sections
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section. If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic or hyperbolic respectively.
Volume
Video: Volume Of A Cylinder
If the base of a circular cylinder has a radius r and the cylinder has height h, then its volume is given by
V = πr2h.
This formula holds whether or not the cylinder is a right cylinder.
Surface area
Video: Surface Area of a Cylinder
Having radius r and altitude (height) h, the surface area of a ......
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.
The term oblong is occasionally used to refer to a non-square rectangle.A rectangle with vertices ABCD would be denoted as ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
Rectangle
Rectangle
Video: Area of a Rectangle
Type
quadrilateral, parallelogram, orthotope
Edges
and
vertices
4
Schläfli symbol
{ } × { }
Coxeter diagram
Symmetry group
Dihedral (D2), [2], (*22), order 4
Dual polygon
rhombus
Properties
convex, isogonal, cyclic Opposite angles and sides are congruent
Characterizations
Video: Understanding Quadrilaterals - Properties of Rectangles
A convex quadrilateral is a rectangle if and only if it is any one of the following:
a parallelogram with at least one right angle
a parallelogram with diagonals of equal length
a parallelogram ABCD where triangles ABD and DCA are congruent
an equiangular quadrilateral
a quadrilateral with four right angles
a convex quadrilateral with successive sides a, b, c, d whose area is .:fn.1
a convex quadrilateral with successive sides a, b, c, d whose area is
Classification
A rectangle is a special case of both parallelogram and trapezoid. A square is a special case of a rectangle.
Traditional hierarchy
A rectangle is a special case of a parallelogram in which each pair .....
Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves.
Video: What's a Venn Diagram, and What Does Intersection and Union Mean?
A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to easily read visualizations; for example, the set of all elements that are members of both sets S and T, S ∩ T, is represented visually by the area of overlap of the regions S and T. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science.
A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.
Example
Sets A (creatures with two legs) and B (creatures that can fly)
This example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. It is important to note that this overlapping region would only contain those elements (in this example creatures) that are members of both set A (two-legged creatures) and are also members of set B (flying creatures.)
Humans and penguins are bipedal, and so are then in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles.
The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly (or both).
The region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles.
In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
Video: How to Calculate Standard Deviation
A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data.
Basic examples
Sample standard deviation of metabolic rate of Northern Fulmars
Logan gives the following example. Furness and Bryantmeasured the resting metabolic rate for 8 male and 6 female breeding Northern fulmars. The table shows the Furness data set.
Furness data set on metabolic rates of Northern fulmars
Sex
Metabolic rate
Sex
Metabolic rate
Male
525.8
Female
727.7
Male
605.7
Female
1086.5
Male
843.3
Female
1091.0
Male
1195.5
Female
1361.3
Male
1945.6
Female
1490.5
Male
2135.6
Female
1956.1
Male
2308.7
Male
2950.0
The graph shows the metabolic rate for males and females. By visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.
The graph shows the metabolic rate for males and females. By visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.
The sample standard deviation of the metabolic rate for the female fulmars is calculated as follows. The formula for the sample standard deviation is
where are the observed values of the sample items, is the mean value of these observations, and N is the number of observations in the sample.
In the sample standard deviation formula, for this example, the numerator is the sum of the squared deviation of each individual animal's metabolic rate from the mean metabolic rate. The table below shows the calculation of this sum of squared deviations for the female fulmars. For females, the sum of squared deviations is 886047.09, as shown in the table.
such ......
ABCD.
.:fn.1
are the observed values of the sample items, 
is the mean value of these observations, and N is the number of observations in the sample.
