Parallelogram Properties

Properties of a parallelogram

A quadrilateral is a plane geometric figure that has four sides and four vertices. If in such a quadrilateral, both the pairs of the two opposite sides are parallel and they are also congruent to each other, then such a quadrilateral is called a parallelogram. (Abbreviated as ||gm). The figure below shows a sample parallelogram.



In the above picture, ABCD is a parallelogram. The two sides AB and DC are parallel to each other. This is indicated by the single arrows. The other two sides AD and BC are also parallel to each other. This is indicated by double arrows on both the sides. Also the length of the side AB is same as that of DC and the length of the side AD is same as that of BC. The four angles of the parallelogram are: Properties of parallelogram:

1. The first property we already stated in the definition of the parallelogram, that the opposite sides are parallel to each other.

2. The opposite sides are also congruent to each other. This we saw in the figure above and its description.

3. The opposite angles are congruent. Thus in the above figure,
4. The adjacent angles are supplementary. Thus in the above figure:
mmmm
5. The diagonal of a || gm divide the || gm into two congruent triangles. This can be shown with the help of the following figure:

In the above figure, ABCD is a || gm. AC is its diagonal. Now if we consider the triangles DAC and BAC, we see that one of the sides AC is common to both the triangles. We already established as the property number 2 of the || gm that the opposite sides are congruent. Therefore the side DC is congruent to AB and DA is congruent to CB. Therefore by the SSS congruency theorem, the two triangles DAC and BAC are congruent. Hence the property 5 of the || gm stands proved.

6. Area of a parallelogram: Since we just established that the diagonal of the parallelogram divides the || gm into two congruent triangles, if we know the area of one of these triangles we can find the area of the || gm by doubling the area of the triangle. Let us now see how to find the area of the || gm.

Consider the || gm ABCD shown in the figure below:

The triangle BDC has the length of base = DC = b and the altitude = h. The area of this triangle is therefore given by the formula:
∆ =(1/2)*base*height
∆ =(1/2)* b*h

Now we already established that the area of the || gm is twice the area of this triangle. Thus  the area of the | | gm ABCD would be:
A=2* ∆
A=2*(1/2)* b*  h
A=b*h

 Properties of normal distribution is a topic of statistics and therefore shall be tackled under a separate article.

The Stem And Leaf Plots

Stem and Leaf Plot Definition:- The stem and leaf plot is used for the presentation of the quantitative data in the graphical formats. It is similar to the histogram by which the shape of the distribution of the data can be found. It is the useful tool which can be used in exploratory data analysis. This plot was popular during the type writer time. In modern computer the machine language is zero and one.

So this technique of layout of the data is obsolete in modern computers. The stem and leaf display is also called as the stem plot. The stem and leaf displays retain the data to at least two significant digits. This display contains two columns which are separated by a vertical line. The left column contains the stem and the right column contains the leaf of the data.

For example in the nub thirty two the stem is left most digits which are three and the leaf is the rightmost digit which is two is the leaf. In numb ten one is the stem and zero is the leaf of the number. In number twenty nine the leftmost digit is two and rightmost digit is nine. The number two is called the stem and the number nine is called the leaf.

To construct the stem-leaf displays the data numbs are arranged in the ascending orders.  The data value may be rounded to a particular place value that can be used as the leaf. The remaining digits to the left of the rounding digit will be used as the stem. Stem and leaf plots can be used to find the range, median, mean and media. Other statistics parameters can also be calculated with this available data.

Irregular Polygon Definition:- Polygon is the plane figure which is bounded by the finite chain of straight line segments and closed in a loop to form a closed chain or circuit. These straight line segments are known as edges or sides. The junction of the two sides is called the vertex or corner. Polygon means a shape which has many sides and angles.

A regular polygon means the shapes which have many sides of equal lengths and many angles which are equal in measurements. Irregular polygons are those in which length of each side is not equal and the measurement of angles is also not equal.  The polygon in which one or more interior angles are greater than one hundred and eighty degrees is called as concave polygon.

The polygon which has only three sides cannot be concave. The convex polygon has opposite properties to the concave. It means one or more angles of the polygon are less than one hundred and eighty degrees.

A line which is drawn through the concave polygon can intersect it more than two places. It is also possible that some of the diagonals lie outside of the polygons.  In convex polygon all diagonal lie inside the polygon.  The area of the concave polygon can be found by assuming it as other irregular polygon.

Spherical Geometry

Spherical Geometry :- The geometrical symmetry is associated with the two dimensional surface of the sphere. It is not Euclidean. The main application of the spherical geometry is in navigation and astronomy. The plane geometry is associated with the point and lines. On the sphere, points are defined as usual sense. The straight lines are not defined as the usual sense. In spherical geometry angles can be defined between great circles, resulting a spherical trigonometry which is differ from the ordinary trigonometry in many respect; the sum of the interior angles of a triangle exceeds one hundred and eighty degrees.

Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter.  To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².

Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.

Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.

In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line. 
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by  {(x₁+x₂)/2  ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment  joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by  {(x₁+kx₂)/(k+1)  ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the  replacing l/m the by k.