Points of concurrency in triangles

There are 3 types of concurrency in triangles.

The centre of these concurrencies are  named   Incentre, circumcentre and orthocenter.

The incenter  helps in forming the in circle in a triangle

The circumcentre is used to circumscribe a triangle.

Incentre:-

The internal bisectors of the three vertical angles are concurrent. This point of concurrency is called Incenter.

I is the Incentre,

I is found  using the formula (ax1+bx2+cx3) / (a+b+c) and  (ay1+ by2+ cy3)/( a+b+c)

 'a' is the side opposite vertex A, 'b' is the side opposite vertex B and  'c' is the side opposite vertex C.

Circumcentre, a Point of Concurrency in Triangles

The perpendicular bisectors of the sides of a triangle are  concurrent. This point of concurrency is called  Circumcenter, and is denoted by the letter S.

The circumcenter is the centre of the circle passing through the 3 vertices  A, B and C

To find S first we find the midpoint of one side of the triangle.  Let us find the midpoint of AB.

Then let us find the slope of AB.

Using slope - one point formula, we can find the circumcenter.

Orthocentre, a Point of Concurrency in Triangles

The altitudes of a triangle are concurrent. This point of concurrency is called orthocenter.  Its symbol is O.

Orthocenter is also found using slope of altitude, one point(coordinate) formula.

For the 2 altitudes we find the equations and then solve the equations.

we must find slope of AD and  use the coordinates of A to find the equation of AD.

Then we must find slope of BE , use the coordinates of B to find the equation of BE'

The solve both equations o AD and BE to get the orthocenter O.

Noun and its Types

Parts of speech are one of the most important lessons in English that every kid needs to understand to get started with the language. There are eight parts of speech – Noun, Pronoun, Verb, Adjective, Adverb, Preposition, Conjunction and Interjection. Noun is the part of speech that refers to a concrete entity. Nouns can be names of people, objects, places or an idea. For example: Ram bought Barbie dolls for his kid. (Ram is a noun). Where is America? (America is a noun). There are two basic types of nouns – Common noun and Proper noun. Further, there are Concrete nouns and Abstract noun.

Common noun: A common noun is the name of a general person, place, thing or an idea. It is not specific to a particular thing. For example: Toys of Fisher Price brand are of great quality. Here, ‘toys’ is a common noun. Cats love drinking milk. Here, ‘cats’ is a common noun.

Proper noun: A proper noun is the name of a specific person, place, thing or an idea. Unlike common noun, it is specific to a particular thing. For example: Tina loves to play with Barbie dolls. Here, ‘Tina’ is a proper noun. Peter is playing with angry birds toys. Here, ‘Peter’ is a proper noun.

Concrete noun: A concrete noun is a type of noun that refers to senses and the ability to hear, touch, feel, smell or taste. For example: Water doesn’t have any color. Here, ‘water’ is a concrete noun. Father bought chair for his kid from Fisher Price brand. Here, ‘chair’ is a concrete noun.

Abstract noun: An abstract noun as the term suggests is a noun form that refers to abstract objects such as ideas or concepts. For example: The judge has given justice. Here, ‘justice’ is an abstract noun. Everyone has their own responsibilities. Here, responsibility is an abstract noun.

These are the basic classifications of nouns. Nouns can also be classified as countable nouns and uncountable nouns. Countable nouns as the term suggests have a singular or plural form while on the other hand uncountable nouns can exist only in singular. Examples of countable nouns are: friends, angry bird toys, houses etc. are countable nouns. Examples of uncountable nouns are: money, bread, etc. These are the different types of nouns that a kid needs to realize in order to get started with the English language.

Consecutive Integers

An integer is a whole number, positive, negative or even 0. There are a number of studies with integers and consecutive-integers are one of those.

Consecutive Integer

An integer which is followed next in order is called the consecutive integer of the given integer. In other words, the difference between these integers is 1. Statistically, in a set of such consecutives, the median of the data is same as mean of the data.

3 Consecutive Integers

Let us understand the concept of consecutive with a few examples. 1, 2, 3 is a set of such since the difference between the integers is constant and the items in the set follow an order. On the other hand 1, 5, 7 is not a set of consecutive as because there is no order is maintained.

Find three Consecutive Integers                                                

We had shown an example of three consecutives as 1, 2, 3. Similarly 3, 2, 1 can also be an example even though the order in the latter case is descending.  For a perfect set of continuous-integers, the difference between these must be 1. To generalize, we can say a set of three consecutive-integers is n, (n + 1), (n + 2), where ‘n’ is any integer. This can be used as a formula to find the three.

Sum of Consecutive Integers                                                

Suppose we take a set of consecutive numbers 5, 6, 7. We can say the sum of these integers is 5 + 6 + 7 = 18. 18 also can be obtained as (3/2)(5 + 7), that is, (half the number of terms)*(first term + the last term). This can be verified with any type of integers and hence this is the formula to find the sum of consecutive-integers.

Problems:

With  these integers, mainly we come across problems like finding the numbers when the sum is given. We use the definition of continuous-integers and solve algebraically. For example if the sum of three consecutive is 24, the numbers can be found as, (n) + (n + 1) + (n + 2) = 24. We get n = 7 and hence the integers are 7, 8 and 9.

Word Problems:

The concept of consecutive-integers is very helpful in real life situations. It can be described through word problems. Suppose you buy fruits for $10 on the first day and your purchase increases by $1 each day. What is the total sum you spent for 5 days?
The fifth day expense is $14 and using the formula the total sum spent is (5/2)($10 + $14) = $60.

Multiplying Scientific Notation

To understand the concept of Multiplying Scientific notation, we should first know what scientific notation is. Scientific notation is expressing numbers in powers of base 10. Scientific Notation is useful because it allows us to track really large and really small numbers in ways that are easy for our brains to write down and to understand. Let us understand by some numbers like 7.102 times 10^2.It is already written in scientific notation already as it is written as a product with a power of ten. So how do we write it just as numeral?

How to Multiply Scientific Notation – We can Multiply Scientific Notation in two ways, there is a slow way and there is a fast way to solve it. The slow way is by simply multiplying, for example if we have 7.102 times 100. That means when we multiply 7.102 with 100 will be 710.200, here firstly we shall multiply the numbers then we shall add the remaining decimal points which gives the correct answer as 710.2.

Now a faster way of Multiplying Scientific Notation is as we see we have only three decimal points and 7.102 times 10 to the 2nd power and when we shift the decimal points by two. Thus the answer would be 710.2. Every time we multiply it by ten, we shift the decimal to the right by one. Let us take another example to understand it better.

Let us say we have 4.5 time 10^4 and this time let us do it in the fast way method. So 4.5 times 10 ^4, here we will shift to the right side after every tenth, so 10^ 4 is nothing but 10000, four times the decimal point will get shifted. So our solution will be 45000.

Let us solve an example with negative exponent to understand the concept in a better way. We have 1.75 times 10 ^-3, this is in scientific notation, and we have to write the numerical value of this. So when we take negative power times ten then should shift the decimal to the left. So this will be 0.00175. We can also call it as the same thing, 1.75 times 10^-3 is equal to 0.00175. Another way to check that we got the right answer is if we count here 1 including the zeros to the right of the decimal should be the same as the negative exponent given.

Whats a Right Angle?

An angle which is exactly 90˚ is called as right angle and is also called as square angle. In geometry, a right angle is an angle, which bisects the angle formed by two halves of a straight line. When the end point of ray is on a line and the adjacent angles are called as right angle. As a rotation, right angle is a quarter of a full circle that is, a right angle corresponds to quarter turn (1/4).

The two perpendicular lines are forms right angle at their point or intersection that angle is called as right angle. In right triangles, the presence of right angle in a triangle is the defining factor, making the right angles basic to trigonometry.

The right angle denoted by a symbol capital alphabet such as “L”. An angle is right angle and expressed by square with the angle as shown in figure. Other angles are having curved angle with a small dot in the field. The units which are used to express the right angles are 90˚, ¼ turn, π/2 radiant, grade or gradian.

Right Angle Formulas
The formula which is used for right angle is,
x1x2 + y1y2 =0
Right angle is formed by perpendicular intersection of two straight lines and is measures 90˚.

Whats a Right Angle?
Right angle is an internal angle and is equal to 90˚. The two edge of the angles are perpendicular to each other are called as right angle. When the special symbol like square box in the angle, it is a right angle. Right angle is exactly equal to 90˚. A right angle can in any rotation or orientation as long as the internal angle is 90˚.

What is a Right Angle in Math?
An angle which measures exactly 90˚ is called as right angle. A right angle is the one quarter of a full rotation. The upper case “L” symbol is used to indicate right angle. In geometry, right angles are one of the most common and interesting angles. It has won the square symbol. When we see the square symbol, we should write the angle is right angle without measuring it.

Definition Right Angle
An angle is formed when the two lines are perpendicular with each other and measure exactly 90˚ is called as right angle. In other words, an angle which equal to half the angle from one end of a line segment to the other is called as right angle. The corner of a piece of paper is an example for right angle. Right angle is expressed by π/2 radiant or 90˚ or ¼ turn. A triangle which contains right angle is called as right triangle. The right triangles should contain only one right angle, since more than one right angle gives more than 90˚ of angle.

Introduction to Ellipse equation

Introduction to Ellipse: 
An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed straight line is always constant and is always less than unity.
The constant ratio is generally denoted by e and is known as the eccentricity of the ellipse.
If S is the focus, ZZ’ is the directrix and P is any point on the ellipse, then by definition SP/PM = e => SP = e. PM.

General Equation of an Ellipse: x^2/a^2 + y^2/b^2 = 1. This is an Ellipse Equation.
Let us understand how to find Equation of Ellipse:
To understand how to Find the Equation of an Ellipse We have, x^2/a^2 + y^2/b^2 = 1, where a > b. ……..(i)So, y  = ± b/a sqrt(a^2 – x^2) ……..(ii) and x = ± a/b sqrt(b^2 – y^2) …….(iii).
We observe the following:
(a) Symmetry: For every value of x an equal and opposite values of y is found.
Similarly for every value of y an equal and opposite values of x is found as well. Thus the curve is symmetric about both the axes.
(b) Origin: The curve doesn’t pass from origin.
(c) Axes Intersection: The curve meets the x axis at y = 0. Putting y = 0 in (iii), we get x = ± a.
So the curve meets x-axis at A (a, 0) and A’ (-a, 0).
Putting x = 0 in (ii), we get y = ± b. So the curve meets y-axis at B(0, b) and B’ (0, -b). (d) Region: If x > a or x < -a, from (ii) we get imaginary values of y.
Therefore there is no part of the curve to the right of A or to the left of A’. If y > b or y < -b, from (iii) we get imaginary values of x.
Therefore there is no part of the curve above B(0, b) or below B’(0, -b). From (ii) we find that at x = 0, y = ± b and as x increases the value of y decreases and y = 0 at x = a.
Therefore the curve is a closed curve. With the help of the above facts and by joining some convenient points on the ellipse the general shape of the ellipse is x^2/a^2 + y^2/b^2 = 1.
Equations of ellipses in other forms: In the equation of the ellipse x^2/a^2 + y^2/b^2 = 1, if a > b or a^2 > b^2, then the major and minor axes lie along x-axis and y-axis respectively.
But if a < b or a^2 < b^2 (denominator of x^2 is less than that of y^2), then the major axis of the ellipse lies along the y-axis and is of length 2b and the minor axis along the x-axis and is of length 2a.
The coordinates of foci S and S’ are (0, be) and (0, -be) respectively. The equations of the directrices ZK and Z’K’ are y = ± b/e and eccentricity e is given by the formula a^2 = b^2 (1 – e^2) or e = sqrt(1 – a^2/b^2).

Ellipse Equation: If the centre of the ellipse is at the point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is (x – h)^2/a^2 + (y – k)^2/b^2 = 1. If we shift the origin at (h, k) without rotating the coordinate axes, then x = X + h and y = Y + k. So, the equation of ellipses in reference to new origin, becomes X^2/a^2 + Y^2/b^2 = 1.

Complementary angles

If a fifth grader asks you, what are complementary angles, you need not stare at him perplexed. This short note would give you enough information to answer that question.
Complementary angles definition is not as complicated as the name sounds. Take a look at the following figure:

We have two angles here. One is 40 degrees and the other is 50 degrees. But that is not important. The important point to be noted here is that 40 + 50 = 90 degrees. That means that the measure of these two angles add up to 90 degrees. With this understanding let us now write the definition of complementary-angles.

Definition: Complementary angles:

Two angles, whose sum is 90 degrees are said to be complementary-angles. The angles may or may not be adjacent angles. For example, see the figure below:

In this figure the two angle measures are 27 degrees and 63 degrees. Sum of these measures is 27 + 63 = 90 degrees. Therefore we can say that these two are a pair of complementary-angles. Examples of other such pair could be: 30 – 60, 45 – 45, 15 – 75, etc. There can be infinite such pairs. When a pair of angles are complementary-angles, we say that one is the compliment of the other, or one compliments the other. A pair of complimentary angles is always acute. That is to say that both the angles have to be acute angles.

Where do complementary angles occur?

Complementary angles are most common in a right triangle. In a right triangle since one angle is 90 degrees, the sum of the other two angles has to be 90 degrees. That is because we know that sum of all the three angles in a triangle should always be 180 degrees. Therefore in a right triangle, the two smaller angles are compliments of each other, or in other words they are complementary-angles.

The concept of complementary angles also finds its application in trigonometry. The sine ratio of an angle is always equal to the cosine ratio of the compliment of that angle. Similarly the tangent ratio of an angle is always equal to the cotangent ratio of the compliment of that angle. Also the secant ratio of an angle would be equal to the cosecant ratio of the compliment of that angle.