Math theory

Isosceles trapezoid

Isosceles trapezoid

In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid.

Video: What is an Isosceles Trapezoid?

Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).

Special cases

Special cases of isosceles trapezoids

Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them.

Another special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid or a trisosceles trapezoid. They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.

Video: Quadrilaterals - Trapezoids, Parallelograms, Rectangles, Squares, and Rhombuses

Self-intersections

Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length.

Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.

Isosceles trapezoid example.png Crossed isosceles trapezoid.png Antiparallelogram.svg
Convex isosceles
trapezoid		 Crossed isosceles
trapezoid

antiparallelogram

Characterizations

If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid (nor, under the definitions given in Wikipedia, is it sufficient, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides); any of the following properties also distinguishes an isosceles trapezoid from other trapezoids:

  • The diagonals have the same length.
  • The base angles have the same measure.
  • The segment that joins the midpoints of the parallel sides is ....

.....

 

 

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Ellipse

Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how "elongated" it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.

Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure below). Ellipses have many similarities with the other two forms of conic sections: parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.

An ellipse (red) obtained as the intersection of a cone with an inclined plane

Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is the above-mentioned eccentricity of the ellipse.

An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.

Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics.

Video: Equation of an Ellipse, Deriving the formula

Definition of an ellipse as locus of points

Ellipse: Definition

Ellipse: definition with director circle

An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

  • An ellipse can be defined using two fixed points, ,  , called the foci and a distance, usually denoted . The ellipse defined with , and   is the set of points such that the sum of the distances  is constant and equal to . In order to omit the special case of a line segment, one assumeshttps://wikimedia.org/api/rest_v1/media/math/render/svg/8f096d24464c8357..."> More formally, for a given , an ellipse is the set

The midpoint  of the line segment joining the foci is called the .....

 

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Transversal

Transversal

In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points.

Video: Parallel Lines, Transversals, and Angles

Transversals play a role in establishing whether two other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, corresponding angles, and alternate angles. By Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Transverzala 8.svg     Transverzala nonparallel.svg Transverzala parallel.svg
Eight angles of a transversal.
(Vertical angles such as and  are always congruent.)		   Transversal between non-parallel lines.
Consecutive angles are not supplementary.		 Transversal between parallel lines.
Consecutive angles are supplementary.

Angles of a transversal

A transversal produces 8 angles, as shown in the graph at the above left:

  • 4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and
  • 4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior, namely α1, β1, γ and δ.

A transversal that cuts two parallel lines at right angles is called a perpendicular transversal. In this case, all 8 angles are right angles 

When the lines are parallel, a case that is often considered, a transversal produces several congruent and several supplementary angles. Some of these angle pairs have specific names and are discussed below: corresponding angles, alternate angles, and consecutive angles.

Video: Figuring out angles between transversal and parallel lines

Corresponding angles

One pair of corresponding angles. With parallel lines, they are congruent.

Corresponding angles are the .......

 

 

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Average

Average

In colloquial language, an average is a middle or typical number of a list of numbers.

Video: Math Help : How to Calculate an Average

Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage sometimes any of these might be called an average value.

Arithmetic mean

The most common type of average is the arithmetic mean.

Video: What is the Arithmetic Mean?

If n numbers are given, each number denoted by ai (where i = 1,2, …, n), the arithmetic mean is the sum of the as divided by n or

 

The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2, 8, and 11, the arithmetic mean is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/3 = 7.

Mode


 

Comparison of arithmetic mean, median and modeof two log-normal distributions with different skewness

The most frequently occurring number in a list is called the mode.

Video: The Mean, Median and Mode Toads

For example, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode. Some authors say they are all modes and some say there is no mode.

Median

The median is the middle number of the group when they are ranked in order. (If there are an even number of numbers, the mean of the middle two is taken.)

Video: Statistics - Find the median

Thus to find the median, order the list according .....

 

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Perimeter

Perimeter

A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference.

Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter.

Formulas

shape formula variables
circle
where  is the radius of the circle and  is the diameter.
regular polygon
where  is the number of sides and  is the distance between center of the polygon and one of the vertices of the polygon.
equilateral polygon
where  is the number of sides and is the length of one of the sides.
rectangle
where is the length and is the width.
triangle
where , and are the lengths of the sides of the triangle.
square/rhombus
where is the side length.
general polygon
where is the length of the -th (1st, 2nd, 3rd ... nth) side of an n-sided polygon.

cardoid  
(drawing with )
 
 
 

The perimeter is the distance around a shape.

Video: What is a Perimeter in Math?

Polygons

.......

 

 

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Right angle

Right angle

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. 

Video: Right angle

The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry.

 

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