Triangles

Triangles

What is a Triangle??

A three sided closed figure is called a Triangle.

Triangles:

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

Types of Triangles:

Classification of Triangles:

Triangles can be classified according to the relative lengths of their sides:

• In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
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• In an isosceles triangle, two sides are equal in length.
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• An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangles to have only two equal sides, whereas others define that an isosceles triangle is one with at least two equal sides.
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• The latter definition would make all equilateral triangles isosceles triangles
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• In a scalene triangle, all sides are unequal
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• The three angles are also all different in measure. Notice that a scalene triangle can be (but need not be) a right triangle.

Triangles are assumed to be two-dimensional plane figures.

A few basic theorems about similar triangles:

• If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
• If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
• If three corresponding sides of two triangles are in proportion, then the triangles are similar.

Two triangles that are congruent have exactly the same size and shape:all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)

Introduction to Trignometry

Introduction to Trigonometry:

Let us first learn how the word Trigonometry took Birth.

The word Trigonometry (from Ancient Sanskrit "trikona" or triangle and "mati" or measure or from Greek trigōnon "triangle" + metron "measure").

Trigonometry is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves.A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

History of Trignometry speaks about Ancient Egyptian and Babylonian mathematicians who lacked the concept of an angle measure, but they studied the ratios of the sides of similar triangles and discovered some properties of these ratios. The ancient Greeks transformed trigonometry into an ordered science.

Here is an example:

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

* The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.

* The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.

* The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.


Hope you like the above example of Trigonometry.
Please leave your comments, if you have any doubt

Polynomials

Polynomials

Introduction:

A Polynomial is an expression of finite length constructed from variables (also known as indeterminates).We can generally find polonomials in a wide variety and a wide range.For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science.

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters). Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x¹, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant.

A polynomial is a sum of terms:

For example, the following is a polynomial:

Now lets learn about:Polynomial Functions:

Polynomial Functions:

A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

Next let us learn about "The Polynomial Equation"

Polynomial Equation:

A polynomial equation is an equation in which a polynomial is set equal to another polynomial.

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x² – y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.

Areas related to Circles

Areas related to Circles:

Perimeter and Area of a Circle — A Review

The distance covered by travelling once around a circle is its perimeter,
usually called its circumference.

In other words, circumference diameter = π
or, circumference = π × diameter
= π × 2r (where r is the radius of the circle)
= 2πr

Let us recall the concepts learnt in earlier classes, through an example.
Example 1 : The cost of fencing a circular field at the rate of Rs 24 per metre is
Rs 5280. The field is to be ploughed at the rate of Rs 0.50 per m2. Find the cost of
ploughing the field (Take π =22/7).
Solution : Length of the fence (in metres) =
Total cost Rate = 5280 = 22024
So, circumference of the field = 220 m
Therefore, if r metres is the radius of the field, then
2πr = 220
or, 2 ×
22
7 × r = 220
or, r =
220 × 7
2 × 22 = 35
i.e., radius of the field is 35 m.
Therefore, area of the field = πr2 =272
× 35 × 35 m2 = 22 × 5 × 35 m2
Now, cost of ploughing 1 m2 of the field = Rs 0.50
So, total cost of ploughing the field = Rs 22 × 5 × 35 × 0.50 = Rs 1925

Areas of Sector and Segment of a Circle:

The circular region enclosed
by two radii and the corresponding arc is called a
sector of the circle and the portion (or part) of the
circular region enclosed between a chord and the
corresponding arc is called a segment of the circle.

Word problem on Time and Work

Some Important Facts about Time and work problems from answers to math problems

• If A can do a piece of work in n days, then A's 1 day work = 1/n

• If A’s 1 day’s work=1/n, then A can finish the work in n days

• If A is thrice as good workman as B,then: Ratio of work done by A and B =3:1. Ratio of time taken by A and B to finish a work=1:3 , 

Let's see an example from 8th grade math worksheets

Question:-

Working together drew and joyce can put a futon frame together in 36 minutes.If it takes drew 60 minutes to put a futon frame together by himself.How long would it take joyce ?

Answer:-

Let us consider that joyce takes x min to complete it.

In 1 min he can do 1/x part of it.

Drew takes 60 min to complete the work

In 1 min he completes 1/60 part of it.

So, working together ,in 1 min they can do [1/x+1/60] part of it

It has given that together they can complete it in 36 min.

So ,together in 1 min they can do 1/36 part of it

So 1/x+1/60 = 1/36

So, x = 90 min

Which means joyce can put the frame together ,by her self in 90 min.

A word problem on sets

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

Here is an example from algebra 1 word problems


Question:-

In a school there are 20 teachers who teach mathematics or physics,of these ,12 teach mathematics and 4 teach both physics and mathematics .How many teach physics

Answer:-

The total number of students =20 (this is we call what is Universal Set U)

Let M be the teachers who teach mathematics

and P be the teachers who teach physics.

the word 'or' gives a clue of union and the word 'and' gives a clue of intersection.

n(M U P)=20

n(M)=12

n(M Ω P)=4

you can see a example of Venn diagram

using the formula

n(M U P)=n(M)+n(P)-n(M Ω P)

20 = 12+n(P)-4

20 = 8+n(P)

subtract 8 on both sides

12 = n(P)

So 12 teachers teach physics

How to find critical points for an equation

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical is a critical value of the function. These definitions admit generalizations to functions of several variables, differentiable maps between R^m and Rⁿ, and differentiable maps between differentiable manifolds.

Lets see a problem from calculus help with probability problems,which explains us
more about this .

Question:-

f(x)=3x⁴-4x³-12x²+6

Differentiating with respect to x

Solution:-

f'(x)=12x³-12x²-24x

we know f'(x)=0 ,so

12x³-12x²-24x=0

12x(x²-x-2)=0

12x(x²-2x+x-2)=0

12x(x-2)(x+1) = 0

x(x-2)(x+1) = 0

Now we use the zero product property ,which states that
if ab=0 ,then a=0 and b=0,this property is from geometry terms and definitions .

so x=0 (x-2)=0 (x+1)=0

x=0 x=2 x=-1

Therefore,x=0,-1,2 are the critical points for y=f(x)