Concepts of geometry - basics(point, line, angle and parallel lines)

Geometry is everywhere around us, in man made structures, in nature, in sports, in art and in lots of more things. In geometry, we have 4 ideas or can also be said as imaginary things i.e. point, line, plane and space and everything else is built and based on the fundamentals of these concepts.

Let's go on with the concepts:

Point
A point is a location is space and it has absolutely no dimensions i.e. no length, no width and no depth. We use a dot, for example; point D or point E, to represent a point.

Line
A line is a set of points that has only 1 dimension, length. The arrowheads shows that it is extending in both the directions and has no end points. Hence it has no fixed length.

• Line segment: It is the part of a line that has a specific length and specific end points. 2 line segments having the same length are called congruent line segments.  
• Ray: It is the part of a line that has only one end point. It extends endlessly in one direction.

Two lines can be related to each other in 4 different ways:

1. Lines that have just one point in common are called intersecting lines.
2. Lines that lie in the same plane but never intersect even if produced endlessly in both directions are called Parallel lines.
3. Two intersecting lines that form a right angle are called Perpendicular lines.
4. Lines that are not in the same plane and do not intersect are called skew lines.

• Plane: The set of points all lying on one surface is called a plane. A wall, surface of a table, floor etc. are all examples of a plane. A plane actually extends endlessly and the surface of a plane has no thickness. At least three points not on the same line are needed to define a plane.

Angle

Two rays(to be precise) that share a common endpoint form an angle. The common endpoint is called the vertex.

Types of angles; 5 main types:

1. Straight angle: An angle measuring 180° is called a straight angle.
2. Acute angle: An angle whose measure lies between 0° and 90° is called an acute angle.
3. Obtuse angle: An angle whose measure lies between 90° and 180° is called an abtuse angle. 
4. Right angle: An angle whose measure is equal to 90° is called a right angle.
5. Reflex angle: An angle whose measure lies between 180° and 360° is called a reflex angle.

The above five are the main ones. But there are many more left.

• Complementary angles: 2 angles are known as complementary angles if the sum of their degree measures 90°.
• Supplementary angles: 2 angles are known as supplementary angles if the sum of their degree measures 180°.
• Adjacent angles: 2 angles having a common vertex and a common side (ray) are called adjacent angles.
• Linear pair: 2 adjacent angles form a linear pair if they are supplementary i.e., their sum is 180°.
• Angles at point: Sum of the angles round a point is 360°.
• Vertically opposite angles: They are the pairs of angles formed by 2 intersecting lines opposite to each other. They are always equal (or congruent).

Parallel lines

Lines in a plane which do not intersect are called parallel lines. A pair of parallel lines are always the same distance (perpendicular distance) apart.

Transversal: A line which intersects 2 or more given lines in distinct points is known as transversal to the given lines. The lines may be intersecting lines. 

How to factorize? ;factorization methods/types and examples

First of all, what is factorization?

It is the method of writing numbers as the product of their factors or divisors.
Now let's go to the methods of factorizing various types of polynomials in detail.

Type 1. Monomial Factors
The basis for multiplication of polynomials is the distributive property of numbers. This property states:

This shows how, a sum of two terms, which have a factor in common, can be expressed as a product. Thus, the factor a in each term on the left side can be taken out as a common factor of the whole expression and we can write ab + ac = a(b + c); here a and b+c are the factors of ab+ac. A factor such as a is a common monomial factor in the terms ab and ac.
Examples;
Consider 21 = 3 * 7; here 3 and 7 are the factors of 21.
Let's take a complex one; 28x³ - 70x².
Now this can be factorized in the following ways:  
14(2x³ - 5x²), x(28x² - 70x), x²(28x - 70),etc.

Type 2. When the Common Factor is a Polynomial
Form: x(a + b) + y(a + b) + z(a + b)
In ab + ac = a(b + c). for all a,b,c.
You may replace a,b,c by expression involving more than 1 term;
For example;
(x + 5)x+(x + 5)y = (x + 5)(x + y)

Type 3. By grouping the Terms
Form: ax + ay + bx + by
Some polynomials with four terms that do not have a common factor in each term, but do have some similar terms, such as 3m + 3n - an - am, can be factorized by grouping. This means grouping in a manner that will express the given polynomial in the form of a binomial with a common binomial factor.
The first step in this process is to separate the given terms into binomial groups having a common factor to each group. Thus, 

3m + 3n - an - am = (3m + 3n) + (- am - an)

When this can be done, each binomial in the expression may be factorized as a binomial with a common monomial factor. That is,

(3m + 3n) + (-am - an) = [(3(m + n)] + [-a(m + n)] = (m + n)(3 - a)

Notice that a polynomial factorable by grouping has 

1. 4 terms

2. can be grouped into two binomials with a common factor in each

3. such that the binomials factor in each group is the same.

Example;
Let's factorize ax + bx + ay + by
sol. ax + bx + ay + by
= (ax + bx) + (ay + by)
= x(a + b) + y(a + b)
= (a + b)(x + y)

Type 4. Factorising Perfect Square Trinomials

Form: a² + 2ab + b²   and a² -2ab + b²

(Trinomials is which two terms are perfect squares and the third term is twice the product of the square roots of these two perfect square terms).

Formulas:
a² + 2ab + b² = (a + b)²
a² - 2ab + b²  = (a - b)²

Example;
Factorise 36x² +60xy + 25y²

= (6x)² + 2(6x)(5y) + (5y)²

= (6x + 5y)²

= (6x + 5y)(6x + 5y)

Type 5. Difference of two squares
Form: a² - b² = (a + b)(a - b)

You must be familiar with products of the following type:

(a+b)(a-b) = a² - b² 
(3x + y)(3x - y) = 9x² - y²
(4x - 7y)(4x + 7y) = 16x² - 49y²

Since in each expression on the right hand side, two perfect squares are separated by a subtraction sign, therefore, each of these expression is referred to as the differences of two squares.

You can reverse this pattern to find a pattern for factorising the difference of two squares.

Since the first term of the difference of two squares is the square of some expression, the first term of each factor must be the square root of the first square.

Similarly, the second term of each factor is the square root of the second square. One factor must have ‘ + ‘ between the terms, and the other must have ‘ – ‘.

Example;

Factorise 4x² - 9y²

Sol. Now this can be split into 3 steps:

1.      This is the difference of two squares, so you find the square root of the first term. (2x)(2x)

2.      Next, you find the square root of the second term. (3y)(3y)

3.      Now fill in the signs. One has a plus sign, the other has a minus sign. Although it does not matter which comes first, you must have one of each. (2x – 3y)(2x + 3y)

squares and it's roots - properties and methods

Square

If a number is multiplied by itself, the product so obtained is called the square of that number.
The square of a number is a number raised to the power of 2.

Perfect square; 
The numbers 1,4,9,16,25,36 are the squares of natural numbers 1,2,3,4,5,6 respectively and are called perfect squares or square numbers. A natural number is called a perfect square or a square number if it is the square of some natural number.













Properties of square numbers:
1. A perfect square is never negative.
2. A square number never ends in 2,3,7 or 8.
3. The number of zeros at the end of a perfect square is always even. e.g.,200*200=40000
4. The square of an even number is always even, e.g. (12)² = 12*12=144
5. The square of an odd number is always odd, e.g. (15)² = 15*15=225

Square root

The square root of a number n is that number which when multiplied by itself gives n as the product.
It is denoted by ' √ '.