Geometry - it's not just a subject in school, in which you need to get a perfect score.It is also a knowledge that often required in life.For example, when building a house with a high roof is necessary to calculate the thickness of the logs and the number of them.It's easy if you know how to find the height of an equilateral triangle.Architectural structures are based on knowledge of the properties of geometric figures.The forms of buildings are often visually resemble them.The Egyptian pyramids, the packets of milk, embroidery, painting and even northern pies - all triangles surrounding the man.As Plato said, the whole world is based on triangles.
isosceles triangle
To make it clearer, as will be discussed below, it is a little bit remember the basics of geometry.
triangle is isosceles if it has two equal sides.They always called side.Side, the dimensions of which are different, is called a base.
Concepts
Like any science, geometry has its basic rules and concepts.They are quite a lot.Consider only those without which our theme will be more clear.
height - a straight line drawn perpendicular to the opposite side.
median - a segment directed from each vertex of the triangle only to the middle of the opposite side.
angle bisector - a ray that divides the angle in half.
bisector of a triangle - this is a direct, or rather, the segment bisector connecting the top of the opposite side.
It is important to remember that the bisector of the angle - is necessarily a beam, and the bisector of the triangle - is part of the beam.
angles at the base
theorem states that the corners are located at the base of any isosceles triangle are always equal.Prove this theorem is very simple.Consider shown isosceles triangle ABC, in which AB = BC.Because of the angle bisector of ABC necessary to HP.We must now consider the two resulting triangles.According to the condition AB = BC, at the side of the HP total triangles and the angles AED and SVD are, because VD - bisector.Remembering the first sign of equality, we can safely conclude that the triangles are considered.Consequently, all the corresponding angles are equal.And, of course, the parties, but will return to this point later.
height of an isosceles triangle
fundamental theorem, which is based on the solution to almost all problems, is: height isosceles triangle bisects and median.To understand its practical sense (or are), you should make support allowance.This requires the cut paper isosceles triangle.The easiest way to do this from an ordinary sheet of notebook in the box.
Fold the resulting triangle in half, aligning the sides.What happened?Two equal triangle.Now check the guesses.Expand received origami.Draw a fold line.With protractor check the angle between the incised line and base of the triangle.What does the angle of 90 degrees?The fact that the line drawn - perpendicular.By definition - height.How to find the height of an equilateral triangle, we understand.Now for the corners at the top.Using the same protractor check the angles formed by the now high.They are equal.So, the height is both bisector.Armed with a ruler, measure the segments into which the height of the base.They are equal.Therefore, the height of an equilateral triangle in half and divides the base is the median.
The proof
Visual aids vividly demonstrates the truth of the theorem.But the geometry - the science quite accurate, therefore, requires evidence.
During consideration of equality of the angles at the base has been proven equal triangles.Recall, WA - bisector, and triangles AED and SVD equal.The conclusion was that the corresponding sides of the triangle and, of course, angles are equal.Hence, BP = SD.Consequently, WA - median.It remains to prove that HP is high.On the basis of equality of triangles under consideration, it turns out that the angle equal to the angle ADV ADD.However, these two angles are related, and are known to give a sum of 180 degrees.Therefore, what they are?Of course, 90 degrees.Thus, HP - is the height in an equilateral triangle, held to the ground.QED.
main signs
- order to successfully meet the challenges should remember the main features of isosceles triangles.They seem to converse theorems.
- If in the course of solving the problem detected by the equality of two angles, then you are dealing with an isosceles triangle.
- If you can prove that the median is also the height of the triangle, safely enclose - isosceles triangle.
- If bisector is the height, then, based on the main features, isosceles triangle belongs to.
- And, of course, if the median and serves as a height, a triangle - equilateral.
Formula 1 height
However, for most of the tasks required to find the arithmetic height value.That is why we consider how to find the height of an equilateral triangle.
Returning to the above figure, the ABC, which has a - sides, in - ground.HP - the height of the triangle, it is designated h.
What is the triangle AED?Since HP - height, then the triangle AED - rectangular leg that you want to find.Using the Pythagorean formula, we get:
AV² = AD² + VD²
determined the expression of HP and substituting its earlier notation, we obtain:
N² = a² - (w / 2) ².
necessary to remove the root:
N = √a² - v² / 4.
If drawn from a root sign ¼, then the formula will look like:
H = ½ √4a² - v².
So is the height in an equilateral triangle.The formula follows from the Pythagorean theorem.Even if we forget the symbolic record, knowing the method to find, you can always bring it.
Formula height
Formula 2 described above is the basic and most commonly used in most of geometrical problems.But she was not the only one.Sometimes it provided instead of a base given angle.When data such as finding a height of an equilateral triangle?To solve such problems it is advisable to use a different formula:
H = a / sin α,
where H - height, towards the base,
a - side,
α - the angle at the base.
If the problem given the angle at the top, in the height of an equilateral triangle is as follows:
H = a / cos (β / 2),
where H - height, lowered to the base ,null,
β - angleat the top,
a - side.
angled isosceles triangle
very interesting property has a triangle, the apex of which is equal to 90 degrees.Consider a right-angled triangle ABC.As in previous cases, WA - height, toward the base.
angles at the base are equal.Calculate their large work will not make:
α = (180 - 90) / 2.
Thus, corners located at the base, always at 45 degrees.Now consider a triangle ADV.It is also rectangular.Find the angle AED.By simple calculations we get 45 degrees.And consequently, the triangle is not only rectangular but also isosceles.The sides AD and VD are the sides and are equal.But
side AD at the same time is a half side of the AU.It turns out that at the height of an equilateral triangle is half the base, but if written in the form of the formula, we get the following expression:
H = w / 2.
should not forget that this formula is only a special case, and can only be used for the right-angled isosceles triangles.
Golden Triangle
Very interesting is the golden triangle.In this figure, the ratio of the side of the base of equal value, called the number of Phidias.Corner located at the top - 36 degrees, with the base - 72 degrees.This triangle admired Pythagoreans.The principles of the Golden Triangle formed the basis of the set of immortal masterpieces.Known to all five-pointed star built at the intersection of isosceles triangles.For many works by Leonardo da Vinci used the principle of the "golden triangle".The composition of "Mona Lisa" is based just on the figures, which create a right pentagram.
Painting "Cubism", one of the works of Pablo Picasso, gaze underlying the isosceles triangles.
