Quadrilaterals Shapes Learning

The word Quadrilateral may look difficult but it is not so. It just means a four sided figure with four vertices (corners). Absolutely, for any figure the number of sides equals the number of angles. It has four angles too. Based on the sides and the angles, we name the figures. The word quadrilateral formed from the words quad (meaning "four") and lateral (meaning "of sides").

The word Quadrilateral doesn't rhyme with the triangles (which has one fewer side and angle) and the pentagons (which has one more side and angle). Hence, we can say Quadrangle by analogy with the Triangle. Similarly, we can also call as "Tetragon" by analogy with the Pentagon.The sides of a quadrilateral may be congruent or non-congruent, may be parallel or perpendicular. The sum of the angles in a quadrilateral is found out by assuming the right angles for each angle. So, the sum of the angles = 90 + 90 + 90 + 90 = 360 degrees.
Lets have a look on the quadrilaterals shapes.

Quadrilaterals Shapes
The quadrilaterals are mainly divided by two types "Regular" and "Irregular". If all the sides and the angles are congruent, it is a Regular Quadrilateral and if they are non-congruent, then they are Irregular Quadrilaterals.Only the Square belongs to Regular quadrilateral, whereas all the other quadrilaterals are Irregular.Now, you may think that Rhombus whose all sides are congruent, why is it not included in Regular quadrilaterals. Well, I agree that the Rhombus has all sides congruent, but the angles ? Are all angles congruent too ? No, the angles in a Rhombus is not congruent.

Making all Quadrilaterals from a Square
If all the sides are equal and all the angles same in a quadrilateral, then it is a Square. Right ?
Let me show you to make all types of quadrilaterals using just a single square.
Draw a square. If you drag one side far away, there becomes a Rectangle. If you drag any two adjacent vertices to any one direction, there becomes a Parallelogram. We can also call a  Parallelogram as a Squashed Rectangle. Pull in one side of a Parallelogram inside the figure (provided the horizontal lines should not get changed), it makes a Trapezoid. If you make all the sides slant in a square, there becomes Rhombus. If you drag one vertex of a Rhombus far away, there becomes Kite.

Conic sections

The conic sections are curves obtained by making sections, or cuts, at particular angles through a cone. First we will discuss about history of conic sections. Conics are amongst the oldest of the curves, and are the oldest math subject studied systematically and thoroughly. The conics had been discovered by Menaechmus, the tutor to the great Alexander. These conics were perceived in order to solve the three problems (a) trisecting an angle, (b) duplicating the cube, and (c) squaring the circle.

If we cut a cone at different angles, then we will obtain different types of conics. There are four different types of conics that we can obtain. That is circle, where the cone is cut at right angle to its axis, ellipse, where the cone is cut at an oblique angle, parabola, where the cone is cut parallel to the generator and finally hyperbola, where a double cone is cut at an angle steeper than the generator.

Let’s discuss about conic sections formulas and conic sections equations. First is circle, the standard formula of circle is  (X^2+y^2=r^2),  where centre is (0,0) and radius is r. second is ellipse, the standard formula of a ellipse is (x^2/a^2+y^2/b^2 =1 a = 1/2) length major axis,b = 1/2 length minor axis, third one is parabola, the standard formula of parabola is 4px=y^2, where p=distance from vertex to focus. And the last one is hyperbola, the standard formula of hyperbola is (x^2/a^2-y^2/b^2=1), where a = 1/2 length major axis, b = 1/2 length minor axis. The general equation for all conic section is (Ax^2+Bxy+Cy^2+Dx+Ey+F=0). And by using the quadratic formula

If (B^2-4AC) <0 circle="circle" curve.="curve." curve="curve" ellipse="ellipse" is="is" no="no" or="or" p="p" point="point" the="the" then="then">If (B^2-4AC) =0, then the curve is a parabola or two parallel lines or a single line or no curve.
If (B^2-4AC)>0, then the curve is hyperbola or two intersecting lines.

The procedure of graphing conic sections, in this part first we will focus on graphing circle. If we cut a circular cone with a plane which is perpendicular to the symmetric axis of cone, then a circle is formed. The intersection line is parallel to the plane generating circle of the cone. A circle means from its center all points are at equal distance.

Now graphing of ellipse, in ellipse long axis is major axis and short axis is minor axis. Intersection point of two axis called vertices. The vertices along horizontal and vertical axis form points. This point along with center will provide method to graph ellipse in standard form.

Introduction to geometry word problems

Geometry 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 the body of the practical knowledge is concerning with the lengths, areas, and volumes. (Source:wikipedia)

In this article we shall discuss about geometry word problems.

Geometry Word Problems Involving Area:

A rectangle is 4 times as extended as it is wide. If the measurement lengthwise is more than 4 inches and the width is less than 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Solution:

Let x = original width of rectangle

Area of the rectangle is

A = l w

Plug in the values from the question

60 = (4x + 4) (x –1)

Use distributive property to remove brackets

60 = 4x2 – 4x + 4x – 4

Put in quadratic formula 4x2 – 4 – 60 = 0

4x2 – 64 = 0

This quadratic written as a difference of two squares

(2x) 2 – (8)2 = 0

Factorize Difference of two squares are

(2x) 2 – (8)2 = 0

(2x – 8)(2x + 8) = 0

Therefore the values for x is

Since x is a dimension, it would be positive.

So, we take x = 4

The question wants the dimensions of the original rectangle.

The width of the given rectangle is 4.

The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the rectangle are 4 and 16.

Geometry Word Problems Involving Perimeter:

 A triangle has a perimeter of 50. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?

Solution:

Let x = length of the equal side

The formula for perimeter of rectangle

P = sum of the three sides

Plug in the values from the question

50 = x + x + x+ 5

Combine like terms

50 = 3x + 5

Isolate variable x

3x = 50 – 5

3x = 45

x =15

The question requires length of the third side.

The length of third side is = 15 + 5 =20

Answer: The length of third side is 20

Practice Geometry Word Problems:

A triangle has a perimeter of 80. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 30

A triangle has a perimeter of 95. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 35

Various types of conics sections

Definition of conic sections: We can define conic section as follows: Consider a double cone. If we have a plane that cuts this double cone, the cross section thus obtained at the intersection of the plane and the cone is called a conic section.

Depending on the angle at which the plane cuts the cone and the position of the plane, we can have mainly 4 different conics: circle, ellipse, parabola and hyperbola.

Conic s have been studied for over 2000 years. Greek mathematician Apollonius studied them intensely. He wrote a book ‘the conic’ that remained a standard work on the topic for eighteen centuries. In the sixteenth century, Galileo declared that the trajectory of a projectile was a parabola. The reflectors in head lights of a car, the speakers in a sound system and the mirrors in a telescope are all in the shape of a parabola.

Parabolic mirrors are used to harness solar energy. In the seventeenth century, Kepler declared that planets revolved around the sun in elliptical orbits. It is because of our knowledge of ellipses that precise predictions of time and place of solar and lunar eclipses is possible. The path of motion of a comet in the solar system is also in the shape of a parabola, ellipse or a hyperbola. Knowledge of conics is extremely useful in such terrestrial sciences as architecture and bridge building.

Thus, the study of conics has been proved very useful in space sciences, mechanics, optics, engineering, architecture and other fields.

How to graph conic sections?
To be able to graph conics, let us look at the following concept. Suppose line l is a fixed vertical line and another line m intersects l in the point V and makes an angle of measure a(0 < a < pi/2) with it. If m is made to rotate around V in such a way that a remains constant, then the surface generated is called a double cone. The point V is the vertex and the line m is a generator of the double cone. The line l is the axis of the double cone.  Then the plane cutting this double cone defines the four conics as follows: (see picture below)