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Showing posts with label triangle. Show all posts
Showing posts with label triangle. Show all posts

3.02.2017

Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

Inverse transformations. We transformed the sum of a line segments to the law of cosines. Mathematics For Blondes.
Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

Measurement of a angle. 0 degree and 180 degrees. Mathematics For Blondes.
Measurement of a angle
It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

If you liked the publication and you want to know more, help me with working on by other publications.

2.27.2017

We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

The Pythagorean theorem and the sum of a line segments. Mathematics For Blondes.
The Pythagorean theorem and the sum of a line segments
In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

Right triangle and two a line segments. Mathematics For Blondes.
Right triangle and two a line segments
How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

Triangle and the law of cosines. Mathematics For Blondes.
Triangle and the law of cosines
Right triangle and the Pythagorean theorem. Law of cosines. Mathematics For Blondes.
Right triangle and the Pythagorean theorem
The law of cosines and sum of two a line segments. Mathematics For Blondes.
The law of cosines and sum of two a line segments
We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

If you liked the publication and you want to know more, help me with working on by other publications.

2.25.2017

Law of sines

The law of sines (sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

The law of sines. Sine law, sine formula, or sine rule. Mathematics For Blondes.
The law of sines
Where:
a, b, c - are the lengths of the sides of a triangle;
α, β, γ - are the opposite angles;
T - are the area of triangle;
R - are the radius of the triangle's circumcircle.

How to use this monster? Use dress-making courses. Cut the law of sines on a part.


Cut the law of sines on a part. Mathematics For Blondes.
Cut the law of sines on a part
Make a necessary formula of two parts. Use properties of proportions.

Example of use of the law of sine. Mathematics For Blondes.
Example of use of the law of sine

8.01.2016

Triangle and rectangle

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
At the first lesson we saw off
Short analysis of trigonometric functions

Lesson 2

Transformations of a triangle to a rectangle


If to combine two rectangular triangles on diagonal so that the rectangle turned out, then the hypotenuse of a triangle will turn into rectangle diagonal, and the parties of a triangle will turn into the parties of a rectangle. The trigonometrical relations of a triangle turn into the trigonometrical relations of a rectangle.

Transformations of a triangle to a rectangle. Mathematics For Blondes.
Transformations of a triangle to a rectangle

Angular symmetry in a rectangle


Diagonal of a rectangle divides a right angle into two trigonometrical angles. The name of a trigonometric function depends on what angle we will take for definition of its numerical value. At the same time the numerical result does not depend on our choice.

Angular symmetry in a rectangle

Names of trigonometric functions and the name of angles possess properties of an angular symmetry. At the same time a symmetry of AND functions of angles are inseparably linked among themselves. If we take the symmetric function and the symmetric angle, then the result will remain invariable. We apply an angular symmetry twice. The similar situation turns out in a plane geometry. If twice to apply a reflecting symmetry, then nothing will change. In algebra analog of an angular and reflecting symmetry is multiplication to minus unit. If an algebraic expression to increase unit by minus twice, the algebraic expression will remain the same.

Reflecting symmetry, the inverse symmetry, angular symmetry, multiplication to minus unit are manifestations of the same law of a symmetry under different conditions.

Considering a symmetry of trigonometric functions, they can be united pairwise in separate groups, each of which expresses particular type of dependence between angles and numbers.

At the following lesson we will consider
Three main types of trigonometric functions

7.25.2016

Trigonometric functions in a rectangle

Published on 7 July, 2016
"The Papers of Independent Authors"
Volume 36 of p. 46-69

Annotation

Representation of trigonometric functions in a rectangle allows to unite algebra, geometry and physics in a single whole.

Short analysis of trigonometric functions

Usually trigonometric functions of plane angle are defined in a rectangular triangle as ratios of the parties of this triangle [a reference to the source in the printing edition].

Trigonometric functions in a triangle. Mathematics For Blondes.
Trigonometric functions in a triangle

If to accept the definitions of trigonometric functions entered by mathematicians in a rectangular triangle, then values of these functions depend only on a ratio of the sizes of the parties of a triangle. The size of angles in a rectangular triangle is in the range of trigonometrical angles. Trigonometric functions have no signs and do not possess periodic. These properties are homocentrism elements.

The homocentrism mathematics is a mathematics in which the result depends on the option of the relative mathematics accepted by us or on our opinion. Striking examples of a homocentrism in mathematician are: division of numbers on positive and the negative, decimal numeration, numbers and inverse numbers, Cartesian coordinate system, etc.

In Cartesian coordinate system trigonometric functions are defined as coordinates of points of a unit circle [a reference to the source in the printing edition].

Trigonometric functions in Cartesian coordinate system. Mathematics For Blondes.
Trigonometric functions in Cartesian coordinate system

This definition is possible only because for any point of a circle of coordinate are used as padding elements for creation of a rectangular triangle. For cross points of a circle and coordinate creation of a triangle is impossible.

For any point of the plane in Cartesian coordinate system trigonometric functions can be defined as the relation of coordinates of this point or the relation of coordinates of a point to distance from a point to the center of a frame. An exception is the cross point of coordinate (the center of a frame) for which trigonometric functions cannot be defined. This fact is congenital defect of Cartesian coordinate system. If it is necessary to define trigonometric functions for the point coinciding with the center of a frame, then the frame needs to be displaced aside.

In Cartesian coordinate system trigonometric functions are the relative and depend on the relative positioning of the plane on which the considered points, and frames are located. Periodicity of trigonometric functions is result of rotation of a piece around the center of a frame. Signs of trigonometric functions depend on the positive direction of coordinate accepted by us. All this result homocentrism of views of trigonometric functions.

Signs of trigonometric functions "+" and "-" serve for orientation in space of Cartesian coordinate system. In mathematical formulas with use of trigonometric functions, the sign "minus" at function automatically changes addition for a subtraction or a subtraction on addition. In formulas it is possible to do without the negative values of trigonometric functions. It will allow to reduce twice amount of values of trigonometric functions, but will increase quantity of formulas.

At the following lesson we will consider
Transformations of a triangle to a rectangle

4.01.2016

Law of cosines in general

To present the law of cosines in general need to go back to the beginning. If the expression for one side of the triangle the length multiplied by the same hand n-times, the equation does not change.

Conversion formulas. Law of cosines in general. Mathematics For Blondes.
Conversion formulas

If you add equal to three sides of the triangle, we obtain the law of cosines in general.

Law of cosines in general. Mathematics For Blondes.
Law of cosines in general


The law of cosines in general terms describes the relationship between the sides and angles of a triangle in a multidimensional space. If the given equation is satisfied, then the triangle is in the Euclidean space. Options description triangle in curvilinear spaces require further study. For the correct application of the law of cosines in solid geometry, spherical triangles side length should be unit of measurement in the same units of measurement, which is measured in planimetrics.

3.28.2016

Degenerate triangle

Finally, we will check the law of cosines to the perimeter in the degenerate triangle. There can be two options. If we decrease the base of an isosceles triangle to zero, we obtain the two overlapping segments. The sum of the angles of this degenerate triangle is 180 degrees.

Degenerate-triangle. Law of cosines to the perimeter. Mathematics For Blondes.
Degenerate triangle

If we combine the upper vertex of the triangle with the base, we get the second type of degenerate triangles. This segment equal to the sum of the other two segments. The sum of the angles are also equal to 180 degrees.

Degenerate triangle. The sum of two segments. Low of cosines. Mathematics For Blondes.
Degenerate triangle

A degenerate triangle - is the lower limit of the application of the law of cosines. After that we will look at the law of cosines in general.

3.24.2016

Isosceles obtuse triangle

Now we will check how the law of cosines in obtuse triangle. For example, consider an isosceles obtuse triangle.

Isosceles obtuse triangle. Law of cosines. Mathematics For Blondes.
Isosceles obtuse triangle


Ups! The perimeter of the triangle may be square roots. The law of cosines is not working? Do not jump to conclusions. The secret is revealed very simple. Let us express the base of the triangle through the sides and see what happens. Double-paste the resulting equation in our result. First time turvy-topsy, second time topsy-turvy.

The perimeter of the triangle. Mathematics For Blondes.
The perimeter of the triangle

The law of cosines to the perimeter works flawlessly. We have completed our review of the degenerate triangle.

P.S. 05/31/16 The онлайн игрыcalculator of triangles confirms the received result.

To calculate a triangle. Mathematics For Blondes.
To calculate a triangle

3.16.2016

Right triangle

Now we will check the law of cosines to the perimeter of the example of a right triangle. At one corner of the right triangle is 90 degrees, the cosine of this angle is zero. Cosine of the other angles are obtained by dividing the length of the hypotenuse to the length of the adjacent side. In general, the law of cosines check looks like.

Right triangle. Law of cosines to the perimeter. Mathematics For Blondes.
Right triangle

Now we substitute in formula for the length of the sides of a right triangle and values of cosines.

Right triangle. Mathematics For Blondes
Right triangle

These values are equal to the perimeter of the triangle. Cosine law allows to check the triangle on the break. That's what happens when one of the sides does not reach the top of the triangle.

Broken triangle. Mathematics For Blondes.
Broken triangle

Equality is performed, but the result is not equal to the sum of the lengths of a broken line or the perimeter of the triangle. You can then proceed to test the law of cosines in the isosceles obtuse triangle.

3.15.2016

Check the law of cosines

Check the law of cosines for the perimeter, we start with an equilateral triangle. All sides of an equilateral triangle are equal. All angles are equal to 60 degrees. The perimeter of the triangle is equal to three times the length of the sides.

Equilateral triangle. Check the law of cosines. Mathematics For Blondes.
Equilateral triangle

Equality is performed. Now we check the law of cosines in a right triangle.

3.14.2016

Law of cosines for perimeter

Carefully look at the proof of the law of cosines and make some corrections.

Analysis of proof. Law of cosines and perimeter. Mathematics For Blondes.
Analysis of proof

Now we can write the law of cosines for the perimeter of the triangle.

Law of cosines for perimeter. Perimeter of triangle. Mathematics For Blondes.
Law of cosines for perimeter

The result is a very simple and beautiful formula that describes the entire triangle. The law of cosines shows a relationship between the angles and the lengths of the sides of a triangle.

The geometry of the law of cosines. Triangle. Mathematics For Blondes.
The geometry of the law of cosines

Law of cosines for perimeter describes the perimeter of the triangle made up of one-dimensional Euclidean spaces. For multidimensional spaces cosine law has a different view.

3.13.2016

Law of cosines

The triangle has three sides and three corners. Appearance cosine theorem depends on the received angles and sides of the triangle symbols. Here's how it looks in Wikipedia.

Law of cosines. The triangle has three sides and three corners. Mathematics For Blondes.
Law of cosines

Three angle of the triangle gives three options for the formula of one triangle. In law of cosines can use a one formula, and three variants of symbols.

Three variants of symbols. Law of cosine for triangle. Mathematics For Blondes.
Three variants of symbols

These two options allow to describe all sides and angles of the triangle. The traditional problems of mathematics we are taught to find one of the triangle elements.

Question: Can one formula with one variant of symbols to describe all the elements of the triangle?

Answer: Yes, you can.
Here's how to do it using the cosine theorem.

The proof of the theorem of cosines in the trigonometric form looks like.

The proof using trigonometry. The proof law of cosines. Mathematics For Blondes.
The proof using trigonometry

If you change the "minus" sign in the "plus" sign, we get the cosine theorem for the perimeter of the triangle.

Law of cosines in general form (in Russian).

3.04.2016

What are the angles of the triangle?

Q: What are the angles of the triangle ABC, if the sum of the angles A and B is equal to 100 degrees, the sum of the angles B and C equal to 120 degrees?
The triangle and the angles. Mathematics For Blondes.
The triangle and the angles
 We have two sum of two angles:

A + B = 100
B + C = 120

The sum of angles in a triangle is 180 degrees.

A + B + C = 180

Substitute in this formula the sum of two angles and find the third angle:

100 + C = 180
C = 180 - 100
C = 80

The second sum of the angles substitute angle C:

B + 80 = 120
B = 120 - 80
B = 40

The first sum of angles substitute angle B:

A + 40 = 100
A = 100 - 40
A = 60

A: The angles of the triangle are equal: A=60 degrees, B=40 degrees, C=80 degrees.

If we substitute in the second sum angles of a triangle, then the solution would be:

A + 120 = 180
A = 180 - 120
A = 60

60 + B = 100
B = 100 - 60
B = 40

60 + 40 + C = 180
C = 180 - 60 - 40
C = 80

The third variant of the decision:

We find the angle A of the first sum.

A + B = 100
A = 100 - B

We find the angle C of the second sum.

B + C = 120
C = 120 - B

Substitute the found angles to the general formula:

A + B + C = 180
(100 - B) + B + (120 - B) = 180
100 - B + B + 120 - B = 180
B - 2B = 180 - 100 - 120
-B = -40
B = 40

A = 100 - 40
A = 60

C = 120 - 40
C = 80

Conclusion: if the problem is solved correctly, the result does not depend on the method of solution.

2.15.2016

Find the angles of an isosceles triangle

Find the angles of an isosceles triangle if the angle opposite the base, at 24 degrees less than the angle at the base of the triangle.

The triangle is isosceles. In an isosceles triangle the angles at the base are equal. The sum of the angles of a triangle is 180 degrees.

Let the angle at the base through X. Then the angle opposite the base, is equal to X-24. We write the formula for the sum of angles of a triangle and find the X:

2х+х-24=180
3х=180+24
х=204/3
х=68


Now we find the third angle:

х-24=68-24=44

Make a check: 2*68+44=136+44=180

Answer: the corners of triangle are equal to 68, 68 and 44 degrees.

4.20.2015

Stupid triangle top view

Maybe it's someone not very pleasant, but stupid triangles does not happen, there are only stupid people. But wits on the Internet are often looking for a stupid triangle, top view which they would very much like to see. Show.

Stupid triangle top view. Obtuse triangle. Mathematics for blondes.
Stupid triangle top view

First, let's talk about the name of this type of triangles. Call similar triangles stupid - a sign of illiteracy. As they say now, "politically correct" will call such triangles "obtuse". All the triangles which have one angle greater than 90 degrees, are obtuse triangle. It's not a shame, is not a defect, just that physique is obtuse triangles. However, each obtuse triangle always has a couple of sharp corners. So, just in case.

Now let's talk about the kinds. No triangles on the kinds and types of top side of the window. Obtuse triangle top looks exactly the same as the bottom. But the side view is not pleasing to the eye - is just a regular interval. From the window you are unlikely to consider anything, but in a notebook at a neighbor's party this triangle you can see from many angles. In this case the obtuse triangle will look quite different, what it looks like from above. Describe such a wonderful transformation is possible by means of projective geometry, descriptive geometry, trigonometry, or poems. Who like more.

1.31.2015

Two angles of a triangle

Consider a very basic problem about two angles of a triangle are known. This problem sounds like this:

Two angles of a triangle are 53 degrees and 57 degrees. Find it the third corner of the triangle.

In any triangle all three angles. That is why the triangle is called. The value of the two angles of the three we know. Now I ask you a couple questions that will help solve this problem.

The first question. What is the sum of the angles of a triangle? This sacred knowledge of mathematics tease "A theorem on the sum of the angles of a triangle." As if they did not call it a law of nature, its essence does not change. Incidentally, the sum of the angles of a triangle belongs to the category of the mathematical knowledge that is easily stored for a long time, but that you never use not awake in their daily lives. Useless knowledge? No, but people use this knowledge is very limited range of professions, such as surveyors.

Sum of the angles of a triangle. The sum of three angles of a triangle is 180 degrees. Mathematics for blondes.
Sum of the angles of a triangle


The second question. If you know that the sum of all the angles of a triangle is 180 degrees, with the arithmetic yourself cope? Here everything is simple. From the sum of the angles of a triangle 180 degrees subtract two prominent corner and get the value of the third angle of the triangle.

180 - 53 - 57 = 70 degrees

I do not want to show here ready-made solution, but ... First, the calculator have a lot of different buttons and accidentally be confused. In such cases, the scientists disappear satellites of Mars. So a complete solution for monitoring, can not hurt. Just check yourself.

Secondly, this is a very good opportunity to do what we do mathematics is strongly not recommended. We are taught to perform tasks with minimal downtime, and possibly without saving intermediate results. Actually, I did. On the one hand, it is correct. On the other hand, it does not give us the opportunity to understand, but what do we actually do?

Personally, I like to consider solving mathematical problems under the microscope in slow motion. Sometimes the impression is that we observe the focus by illusionist and all the secrets of the focus immediately crawl out. Let's look at the detailed solution of this problem on two well-known corners of the triangle and one unknown. Here's how it looks.

Two angles of a triangle. Solution of the problem. Mathematics for blondes.
Two angles of a triangle

And so. Someone measured the angles in a triangle is real. The measurements were performed only for the two corners. Man in high school and knows that the third angle can be simply calculated. This is the condition of the problem. Now, a detailed description of the meaning of the decision and the action carried out by us.

1. Write a law that establishes a relationship between the angles of a triangle, in algebraic form. I have already said that in mathematics it is called "A theorem on the sum of the angles of a triangle." The geometric shape of this law is shown in the first picture.

2. Transform the algebraic form of the law on the corners of a triangle to solve our specific problem.

3. Enter this formula in the data from the task ahead of us. Pass from the algebraic form to the physical.

4.Analiziruem physical model for solving the problem. Mathematical apparatus introduced the decimal system of numbers, other notations are absent. The physical device is represented by a measure of the degree angles, other angle units available. Only under these conditions we can perform addition and subtraction.

5. Go to the mathematical model of the physical problem and perform mathematical operations with numbers using a calculator, a sheet of paper or in your mind.

6. Get ready solution to the problem in physical form.

Here's a novel in verse about I turned to a very simple task. The accuracy of the description of this literary opus does not claim because the school did not teach me this, had to invent on the fly. All the described actions we perform automatically, without going into detailed explanations. I agree with mathematicians that stupid every solution of the problem in as much detail paint. But even more stupid stupid to perform the actions that you teach. In this case, the formation is converted into a conventional animal training.

1.20.2015

Right triangle trigonometry

I will not tell you the rules and definitions of trigonometric functions on a right triangle. Mathematics is happy to do without me. I'll just show you a picture, which shows a right triangle trigonometry.

Right triangle trigonometry. Mathematics For Blondes.
Right triangle trigonometry

The top row shows who is who in the trigonometric zoo. Sine and cosine of the angle alpha - a relationship of legs to the hypotenuse. Tangent and cotangent - a relationship of legs. With hypotenuse usually no problem, and it is located opposite one of the right angle. But the legs as much as two and they are different. One is located in front of the angle alpha is called opposed (the picture side a). Another gently pressed against the corner called alpha and an adjacent (on the picture side b). Now, looking at the picture, you can easily formulate the definition of trigonometric functions on a right triangle.

The bottom row of images shows how to find the sides of a right triangle, if we know one side and the angle alpha. Known side is green. Using this side and trigonometric functions, you can easily find the other two sides of a right triangle.

Spin this picture as you like, turn face down and look up to the light - trigonometric dependence in a right triangle from it are not changed.

Right triangle trigonometry. Spin this picture. Mathematics For Blondes.
Spin this picture

This picture you can come in handy in the future, the study of physics, theoretical mechanics, in engineering calculations. By the time you've forgotten how to define and use trigonometric functions in a right triangle.

And the saddest thing in the end. If you teach mathematics to use mathematics, these pictures you drew to themselves for a few minutes, without any textbooks. After all this is done simply elementary.

7.08.2012

Trigonometric functions determination through a triangle

Trigonometric functions can be determined on a rectangular triangle. On a picture you can see determinations of all trigonometric functions through the elements of rectangular triangle.

Trigonometric functions determination through a triangle. Mathematics for blondes.
It is another icon for hanging out on a wall and overlearnings of text. We this we will not engage in. We will look better, as possible to combine a rectangular triangle and circumference. For this purpose we will combine two pictures: circumference from classic determination of trigonometric functions and our triangle. We will place a triangle next to a circumference. That is it looks.

Triangle and circumference. Mathematics for blondes.
As see, pictures are practically identical, and component elements are named differently in them. Identically mark only corner alpha and point of "B". Now we will impose a triangle straight on a circumference. All graphic images accepted for a circumference we will save, and a rectangular triangle we will underline red lines on the inside (type we will lead around the contour of triangle lipstick).

A triangle is in a circumference. Mathematics for blondes.
As be obvious from a picture, the hypotenuse of rectangular triangle grows into the radius of circumference, cathetuses become equal to the coordinates of point. It as in a human language - a the same concept in different languages is designated by different words. This distinction in a pronunciation does not give to us to understand foreign languages. Approximately the same takes place in mathematics. Some consider determination of trigonometric functions on a rectangular triangle primitive. If you want to understand mathematics, will memorize the following: there are not primitive things in mathematics. There are primitive creatures counting itself very clever. Exactly desire to seem more clever than other, resulted in that mathematicians almost nothing is understood in mathematics.

Specially I will quote a phrase from Russian-language Wikipedia, where talked about determination of trigonometric functions on a triangle: "This determination has some pedagogical advantage, because does not require introductions of concept of the system of coordinates, but also such large defect, that it is impossible to define trigonometric functions even for obtuse angles that must be known at the decision of elementary tasks about amblygons". Honestly speaking, like thick ignorance of mathematicians simply shocks me. What triangle are trigonometric functions determined for? Correctly, for RECTANGULAR. Let though one mathematician will show to me RECTANGULAR TRIANGLE With OBTUSE ANGLE. I swear to you, as soon as I will see this geometrical miracle, I here will invert the portrait of tangent heels over head and I will hang oneself on a cosine. Not a single mathematician understands really, that the decision of "elementary tasks about amblygons" through trigonometric functions is ALWAYS taken to breaking up of one amblygon on two rectangular triangles?

Now I will show the most primitive example of existence of trigonometry you in life. More primitive, probably, is not already.

Trigonometry is in life. Mathematics for blondes.
Look at a photo - how many human bodies serenely stationed oneself on the horizontal plane of rock, at the same time there is nobody on a vertical plane. Why? And because trigonometric functions for perpendicular direction have quite another values. For us with you basic sense of trigonometric functions is that exactly trigonometric functions determine our possibilities.

2.02.2011

Types of triangles

Types of triangles differ on clothes. Here when you look at the person, you, first of all, estimate it clothes and at once much becomes clear to you - what dirty trick from this person can be expected. Precisely as well at triangles. However, triangles have only two clothes which can be estimated simultaneously are corners and the parties. On a picture of a triangle it looks here so:

Types of triangles. Rectangular  triangle, right triangle, obtuse and acute triangle. Mathematics for Blondes.
Types of triangles.
The very first triangle on a picture - a right triangle. It is called so because this triangle has one right angle. Two right angles at a triangle does not happen - remember the triangle theorem? Correctly, if two corners take away to themselves on 90 degrees, to the third corner nothing does not remain. And what can be a triangle with two corners? There are no two-coal triangles. Everything, the theorem of a right angle in a right triangle for blondes is proved!

Types of triangles. Scalene, equilateral and isosceles triangle. Mathematics for Blondes.
Types of triangles.
It is possible to name a right triangle a triangle of Pythagor, on it Pythagor proved the well-known Pythagorean theorem. Let for Pythagor a right triangle also remains. Further on a picture at us the acute triangle is drawn. In it all three corners sharp, as female uvulas. Means, this triangle female. The top part of portrait gallery of triangles comes to an end with an obtuse triangle. This triangle has one big corner, and this corner stupid. It is a man's triangle. No, not because men stupid! No! Think. If there is a female triangle, means should be and man's. And all other triangles are already disassembled! There was only this... Here!

In the bottom part of a cloth of a brush of the unknown artist of the XXI-st century we see triangles which differ with lengths of the parties. To the first in the bottom number there is an equilateral triangle - a democracy symbol. Because all three parties at an equilateral triangle of equal length so - they are equal among themselves.

Further the isosceles triangle is drawn. A symbol of grace and beauty of women. At this triangle two parties have identical length. You imagine the woman with hips of different length? Here, and I do not represent. Why an isosceles triangle named so? In honour of blondes of whom mathematicians dreamt, drawing a pencil a triangle that under a hand has got. After all the majority of mathematicians there were no Leonardo's da Vinci's. Instead of fine Mona Liza's at them them isosceles triangles turned out.

The latest triangle - a scalene triangle - a symbol of geometrical tortures. And you thought, it is connected with erudition of this triangle? As though not so! At this triangle the teacher wants to know all three parties of different length and length of each of the parties of a triangle! After tortures by such triangle even Pythagor has not sustained and has told the theorem. In general, the scalene triangle is the most terrible nightmare of pupils on an extent already more than two thousand years.

Triangle names and pictures - it here. Types of triangles and their name.

1.30.2011

Triangle

The triangle is the elementary polygon. Such definition of a triangle is given by Wikipedia. We will not argue with collective reason, we will try to judge by own strength. The triangle is a geometrical figure which consists of three tops and three parties. As appears from the name "triangle", this geometrical figure has three corners. How the triangle looks? Here a triangle photo in all its beauty.

Triangle, corners, parties of triangle. Mathematics  for blondes.
Triangle

Despite such simple-minded appearance of a triangle, I very much doubt correctness of the statement of Wikipedia that the triangle is the elementary polygon. It is too much at a triangle of any gadgets. Has specially looked at the mathematical directory and here acknowledgement of my words: to a triangle it is devoted 6 (six!) the pages, to all quadrangles together taken, only 5 (five) pages. External simplicity of a triangle at all does not mean simplicity mathematical.

And so, we will start to assort a triangle on stones. Three points which are not lying on one straight line, form a triangle. These points are called as triangle tops. The request not to confuse to mountain tops is absolutely another. The triangle has three tops which are designated by the big Latin letters A, B, C (it is a surname at tops such). You ask, than these Latin letters differ from Russian letters And, In, With? A family tree at these letters different, and consequently also an order of an arrangement of these letters in the alphabet.

Between triangle tops there are triangle parties. These are such equal paths on which it is possible to run across from one top of a triangle to other top. The triangle parties also are inviolable frontiers of a triangle. In these borders there is a triangle area. Everything that is outside behind these borders, into the triangle area does not enter. Along these borders frontier guards with dogs go and watch, that another's area has not got into the triangle area, and the triangle area has not run away abroad from such good life in a triangle. The triangle parties by small Latin letters a, b, c are designated.

Just the same small Latin letters a, b, c the length of these parties of a triangle is designated. The strict boundary heads after all need to know, how many kilometres poor boundary dogs have run? How many kilometres passed unfortunate frontier guards are, for some reason, chiefs never interests. When speech comes about length of the parties of a triangle, probably other designation - two big Latin letters with two vertical sticks: |AB |, |BC |, |AC |. In this case the designation of length of the party of a triangle undertakes on a surname of tops of a triangle between which there is this party (look a photo triangle). In full conformity with rules of mathematical bureaucracy it is possible to write down:

a = |BC|

b = |AC|

c = |AB|

From here the first law of a triangle for blondes is very easily deduced: the length of the party of a triangle is designated by two big Latin буковками or one small буковкой, that which is not present among big буковок.

It is very logical to assume that triangle corners also have the designations. Each corner of a triangle has cosy settled down in triangle top between two parties. Designate triangle corners small letters of the Greek alphabet α (alpha), β (beta), γ (gamma). The sum of all corners of a triangle is equal 180 degrees. Such to itself democracy of a triangle: if you a corner also want to be more, select at other corner and use. All as in life. Therefore corners in a triangle meet the different: stupid and pompous (you such perfectly know) near to thin and graceful (blondes). Democracy of a triangle in the mathematician is called "the triangle Theorem" and it sounds so: the sum of all corners of a triangle equals hundred eighty degrees. In mathematical symbols the triangle theorem looks so:

α + β + γ = 180°

Depending on a kind of the corners which have formed Open Company "Triangle Ltd.", triangles differ on appearance. It is such geometrical dress-code for triangles. But about it we will talk next time.

Here you will find answers for such questions: the sum of all corners on a triangle.