Geometry
A branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes.
Thales
- Thales of Miletus was the one who began early Greek geometry in the sixth century B.C. He is noted as one of the first known to indulge himself in deductive methods in geometry. His credited elementary geometrical findings resulted from logical reasoning rather than intuition and experiment. He insisted that geometric statements be established by deductive reasoning
rather than by trial and error. - Thales went to Egypt and studied with the priests, where he learned of mathematical innovations and brought this knowledge back to Greece. Thales also did geometrical research and, using triangles, applied his understanding of geometry to calculate the distance from shore of ships at sea. This was particularly important to the Greeks, whether the ships were coming to trade or to do battle.
- While Thales was in Egypt, he was supposedly able to determine th
e height of a pyramid by measuring the length of its shadow when the length of his own shadow was equal to his height. - The Greeks named their paper explorations "geometry" for "earth measure," in honor of the Egyptians from whom the knowledge came.
Thales is credited with the following five theorems of geometry:
1.A circle is bisected by its diameter.
2.Angles at the base of any isosceles triangle are equal.
3.If two straight lines intersect, the opposite angles formed are equal.\
4.If one triangle has two angles and one side equal to another triangle, the two triangles are equal in all respects.
5.Any angle inscribed in a semicircle is a right angle. This is known as Thales' Theorem.
Pythagoras
Distance formula
The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry.
Distance Formula: Given the two points (x1, y1) and (x2, y2), the distance between these points is given by the formula: |
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Here's an example:
Find the distance between points (-2,3) and (-4,4) I just plug the coordinates into the Distance Formula: |
Undefined terms
Is a term or word that doesn’t require further explanation or description. It already exists in its most basic form.
Point – A point specifies an exact location.
- In geometry, a point has no dimension (actual size). Even though we represent a point by a dot, the point has no length, width, or thickness.
- In Analytical Geometry a point represents an ordered pair.
Line – In geometry line has no thickness but its length extends in one dimension and goes on forever in both directions.
1) A line can be straight or curve
2) Straight line is produce by a moving point travelling in one direction
3) Curve line is produce by a moving point changing its direction.
4) Two points determine a line.
5) Through any two points is exactly one line.
6) Given a line and a plane, there exists at least one point in the plane that is not on the line.
7) Two Different lines intersect in at most one point.
8) On a line there is a unique distance between two points.
1) A line can be straight or curve
2) Straight line is produce by a moving point travelling in one direction
3) Curve line is produce by a moving point changing its direction.
4) Two points determine a line.
5) Through any two points is exactly one line.
6) Given a line and a plane, there exists at least one point in the plane that is not on the line.
7) Two Different lines intersect in at most one point.
8) On a line there is a unique distance between two points.
KINDS OF LINE
Parallel Lines, Intersecting Lines, and Asymptotic Lines.
Intuitive concept
There are a few basic concepts in geometry that need to be understood, but are seldom used as reasons in a formal proof.
Collinear Points = 0points that lie on the same line.
Coplanar points - points that lie on the same plane.
Opposite rays - 2 rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line and/or a straight angle (180°:).
Parallel lines - two coplanar lines that do not intersect
Skew lines - two non-coplanar lines that do not intersect.
Collinear Points = 0points that lie on the same line.
Coplanar points - points that lie on the same plane.
Opposite rays - 2 rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line and/or a straight angle (180°:).
Parallel lines - two coplanar lines that do not intersect
Skew lines - two non-coplanar lines that do not intersect.
Linear Equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. a first degree equation.
2x-1=7
2x+3y=4
2x-1=7
2x+3y=4
Special Products
Products that most of the time appears in algebra. We can exactly see the answer in the equation.
a) Square of Binomial - (a + b)2 this results into a Perfect Square Trinomial (PST) = (a + b)2 = a2 + 2ab + b2
b) Product of Sum and Difference of two Binomials - (a+b)(a-b) this results into Difference of Two Squares = a2 - b2
c) Square of Trinomial - (a+b+c)2 this results in a^2 + b^2 + c^2 + 2ab + 2ac + 2bc form.
d) Product of Binomials - (a+b)(c+d) this results in ac+[bc+ad]+bd form.
a) Square of Binomial - (a + b)2 this results into a Perfect Square Trinomial (PST) = (a + b)2 = a2 + 2ab + b2
b) Product of Sum and Difference of two Binomials - (a+b)(a-b) this results into Difference of Two Squares = a2 - b2
c) Square of Trinomial - (a+b+c)2 this results in a^2 + b^2 + c^2 + 2ab + 2ac + 2bc form.
d) Product of Binomials - (a+b)(c+d) this results in ac+[bc+ad]+bd form.
Factoring
- Factoring - Finding what to multiply to get an expression; reverse of special product
1. The first term and the last term are perfect squares
2. The coefficient of the middle term is twice the square root of the last term multiplied by the square root of the coefficient of the first term.
When we factor a perfect square trinomial, we will get
(ax)2 + 2abx + b2 = (ax + b)2
The perfect square trinomial can also be in the form:
(ax)2 – 2abx + b2
In which case it will factor as follows:
(ax)2 – 2abx + b2 = (ax – b)2
Examples:
x2 + 8x + 16
= x2 + 2(x)(4) + 42
= (x + 4)2
4x2– 20x + 25
= (2x)2– 2(2x)(5) + 52
= (2x – 5)2• 4x2+12x+9 = (2x+3)2
• 16x2+40x+25 = (4x+5)2
• 9x2-42x+49 = (3x-7)2
- Factoring Difference of Two Square
• x2+y2 = non-factorable
• x2-y2 = (x+y)(x-y)
• 49x-25 = (7x+5x)(7x-5x)
• 81x2-1/4y2 = (9x+1/2y)(9x-1/2y)
• x2y – a10 = (xy3+a5)( xy3-a5)
b4 b2 b2
- Factoring General Trinomial
Example:
• x2+5x-14
sum product (+7 and -2)
= (x+7)(x-2)
• x2-11x+30
sum product (-6 and -5)
(-6)(-5) = 30 (-6)+(-5) = 11
= (x-5)(x-6)
• 2x2-5x-3
-6 (-6 and 1)
2x2-6x+x-3
2x(x-3)+1(x-3)
= (x-3)(2x+1
• 8x2+2x-3
-24 (6,-4)
8x2+6x|-4x-3
2x(4x+3)-1(4x+3)
= (4x+3)(2x-1)
Factoring Square of Trinomial
(4a-
Quadratic Equation
Quadratic Equation (2nd Degree Equation)
• An equation in the form: ax2 + bx + c = 0
where: a is not equal to zero ( a ≠ 0 )
a, b, c must be an element of a real number.2x2 + 5x + 3 = 0 In this one a=2, b=5 and c=3
6x² + 11x – 35 = 0
2x² – 4x – 2 = 0
-4x² – 7x +12 = 0
20x² –15x – 10 = 0
x² –x – 3 = 0
• An equation in the form: ax2 + bx + c = 0
where: a is not equal to zero ( a ≠ 0 )
a, b, c must be an element of a real number.2x2 + 5x + 3 = 0 In this one a=2, b=5 and c=3
6x² + 11x – 35 = 0
2x² – 4x – 2 = 0
-4x² – 7x +12 = 0
20x² –15x – 10 = 0
x² –x – 3 = 0
Incomplete Quadratic Equation
A quadratic equation lacking of the linear part or the constant part
2x² – 64 = 0
x² – 16 = 0
9x² + 49 = 0x² – 7x = 0
2x² + 8x = 0
-x² – 9x = 0
2x² – 64 = 0
x² – 16 = 0
9x² + 49 = 0x² – 7x = 0
2x² + 8x = 0
-x² – 9x = 0