Geometry Parallelograms

Introduction for geometry parallelograms:

In geometry, parallelogram is a shape that has four sides where the opposite sides are parallel to each other. The main concepts of the parallelograms are,

The opposite angles are equal

The opposite sides are equal in its length and are parallel to each other.

Now we are going to see about the geometry - parallelograms and its problems.


Problems for geometry parallelograms:

Example 1:

Find the sides of the parallelogram having 10 cm which is the smaller side of the parallelogram. The longest side is 3 times the smallest side of the parallelogram.

Solution:

The smallest side of a non regular parallelogram = 10 cm (known)

The longest side of the parallelogram will be 10 × 3 = 30 cm.

This is due the opposite sides are equal in the parallelograms

Thus the other two sides are 10 cm and 30 cm respectively.

The irregular quadrilaterals sides, parallelograms = 10 + 30 + 10 + 30 = 80 cm.

Example 2:

Determine the sides of the parallelogram having 15 cm which is the smaller side of the parallelogram. The longest side is 5 times the smallest side of the parallelogram.

Solution:

The smallest side of a non regular parallelogram = 15 cm (known)

The longest side of the parallelogram will be 15 × 5 = 75 cm.

This is due the opposite sides are equal in the parallelograms

Thus the other two sides are 15 cm and 75 cm respectively.

The irregular quadrilaterals sides, parallelograms = 15 + 75 + 15 + 75 = 180 cm.

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More problems for geometry parallelograms:


Example:

Determine the area of parallelogram where the base and height of the parallelogram are 12 cm and 20.

Solution:

Given data is the base, b =12 cm and the height, h =20 cm

We know the formula for the area of parallelogram and given as,

Area of the parallelogram = b × h

Substitute the value of b and h,

Area of parallelogram = 12 × 20

= 240 cm^2

Therefore the area of a parallelogram is 240 cm^2

Geometry Concurrent Lines

Introduction to Geometry concurrent lines

Concurrence
Definition Of concurrent lines
Examples of concurrent lines
Concurrence in Triangle
Concurrence in Circle
Concurrence

The phenomenon when  multiple lines meet at a point is known as concurrence.


When two or more  lines in a plane intersect at a common point then they are said to be concurrent lines.

Examples of geometry concurrent lines

Altitudes of a triangle are concurrent lines
Angular bisector of a triangle are concurrent lines
Perpendicular bisectors of a triangle are concurrent lines
The medians of a triangle are concurrent lines
The diameters of a circle are concurrent lines

Geometry concurrent lines in a triangle


Incenter is the point of concurrence of the angular bisector of a triangle , therefore  the angular bisectors of a triangle are concurrent lines. Angular bisectors are the lines which divide each angle of a triangle in two equal angles they meet at in center.
Circumcenter is  the point of concurrence of perpendicular bisectors of a triangle, therefore perpendicular bisectors of a triangle are concurrent lines. Perpendicular bisectors of a triangle are the lines which divide each side in two equal parts they meet at the circumcenter .


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Orthocenter  is the point of concurrence of altitudes of a triangle, therefore  altitudes of a triangle are concurrent lines. Altitudes are the perpendicular from each vertex of a triangle to the opposite sides, they meet at the ortho center.
Centroid is the point of concurrence of medians of a triangle , therefore medians of a triangle are concurrent lines. Medians are the lines joining the vertex to the mid point of opposite sides, they meet at centroid.
Geometry concurrent lines in a circle.

Center of a circle is the point of concurrence of all the diameter, therefore all the diameters of a circle are concurrent lines . Diameter of a circle is the line joining  two points on the circumference passing through the center .

Three Different Types of Geometry

Introduction:

A non-Euclidean geometry is learning of figures and structure that do not chart straight to any n-dimensional Euclidean system, describe by a non-vanishing Riemann curve tensor. Examples of non-Euclidean geometries contain the hyperbolic and elliptic geometry, which are difference with a Euclidean geometry. The necessary difference among Euclidean and non-Euclidean geometry is the character of parallel lines.


Behavior of lines


Three different types of geometry method to explain the difference connecting these geometries is to think double directly lines indefinitely extensive in a two-dimensional level surface that are together vertical to a three line types:

In Euclidean geometry the position remain at a stable distance starting each other, and are well-known as parallels.
In hyperbolic geometry they "curve away" starting each other, rising in distance as one moves further from the position of intersection through the general perpendicular; these lines are frequently called ultra parallels.
In elliptic geometry the positions “curve toward" each extra and finally intersect.


Models of non-Euclidean geometry


Let us see about three different types of  geometry,

Elliptic geometry

The simplest type for elliptic geometry is a globe, anywhere lines are "great circles" (such as the equator or the meridians on a globe, and points reverse each other are recognized (considered to be the equal).In the elliptic type, for some certain line l and a point A, which is not on l, all position throughout A will intersect l.

Hyperbolic geometry

The pseudo globe has the suitable curve to model a section of hyperbolic space, and in a second document in the similar year, defined the Klein model, the Poincaré disk type, and the Poincaré half-plane type which type the total of hyperbolic space, and old this to explain three different types of geometry that Euclidean geometry and hyperbolic geometry be equip reliable, so that hyperbolic geometry was reasonably constant if and simply if Euclidean geometry.


Their Relationship to Each Other


Let us see about three different types of  geometry,

The different geometries are divided and connected to single another in different ways. The non-Euclidean geometries are closely similar to the geometry of Euclid, but that Euclid's postulate concerning analogous lines is replace and all theorems depending on this assume are changed therefore both Euclidean and non-Euclidean geometry are models of metric geometry, in which the length of line division and the volume of position may be careful and compared.