Math, which I like. Isosceles triangle. Detailed theory with examples

In which two sides are equal in length. Equal sides are called lateral, and the last unequal side is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

Terminology

If a triangle has two equal sides, then these sides are called sides, and the third side is called the base. The angle formed by the sides is called vertex angle, and angles, one of whose sides is the base, are called corners at the base.

Properties

  • Angles opposite to equal sides isosceles triangle, are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.
  • The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.

Let a- the length of two equal sides of an isosceles triangle, b- length of the third side, h- height of an isosceles triangle

  • a = \frac b (2 \cos \alpha)(a corollary of the cosine theorem);
  • b = a \sqrt (2 (1 - \cos \beta))(a corollary of the cosine theorem);
  • b = 2a \sin \frac \beta 2;
  • b = 2a\cos\alpha(projection theorem)

The radius of the incircle can be expressed in six ways, depending on which two parameters of the isosceles triangle are known:

  • r=\frac b2 \sqrt(\frac(2a-b)(2a+b))
  • r=\frac(bh)(b+\sqrt(4h^2+b^2))
  • r=\frac(h)(1+\frac(a)(\sqrt(a^2-h^2)))
  • r=\frac b2 \operatorname(tg) \left (\frac(\alpha)(2) \right)
  • r=a\cdot \cos(\alpha)\cdot \operatorname(tg) \left (\frac(\alpha)(2) \right)

Angles can be expressed in the following ways:

  • \alpha = \frac (\pi - \beta) 2;
  • \beta = \pi - 2\alpha;
  • \alpha = \arcsin \frac a (2R), \beta = \arcsin \frac b (2R)(sine theorem).
  • The angle can also be found without (\pi) And R. A triangle is divided in half by its median, and received The angles of two equal right triangles are calculated:
y = \cos\alpha =\frac (b)(c), \arccos y = x

Perimeter An isosceles triangle is found in the following ways:

  • P = 2a + b(a-priory);
  • P = 2R (2 \sin \alpha + \sin \beta)(a corollary of the sine theorem).

Square the triangle is found in the following ways:

S = \frac 1 2bh;

S = \frac 1 2 a^2 \sin \beta = \frac 1 2 ab \sin \alpha = \frac (b^2)(4 \tan \frac \beta 2); S = \frac 1 2 b \sqrt (\left(a + \frac 1 2 b \right) \left(a - \frac 1 2 b \right)); S = \frac 2 1 a \sqrt \beta = \frac 2 1 ab \cos \alpha = \frac (b^1)(2 \sin \frac \beta 1);

See also

Write a review about the article "Isosceles Triangle"

Notes

Excerpt characterizing the Isosceles triangle

Marya Dmitrievna, although they were afraid of her, was looked at in St. Petersburg as a cracker and therefore, of the words spoken by her, they noticed only a rude word and repeated it in a whisper to each other, assuming that this word contained all the salt of what was said.
Prince Vasily, Lately especially often forgetting what he said, and repeating the same thing a hundred times, spoke every time he happened to see his daughter.
“Helene, j"ai un mot a vous dire,” he told her, taking her aside and pulling her down by the hand. “J”ai eu vent de certains projets relatifs a... Vous savez. Eh bien, ma chere enfant, vous savez que mon c?ur de pere se rejouit do vous savoir... Vous avez tant souffert... Mais, chere enfant... ne consultez que votre c?ur. C"est tout ce que je vous dis. [Helen, I need to tell you something. I have heard about some species regarding... you know. Well, my dear child, you know that your father’s heart rejoices that you... You endured so much... But, dear child... Do as your heart tells you. That's all my advice.] - And, always hiding the same excitement, he pressed his cheek to his daughter's cheek and walked away.
Bilibin, who has not lost his reputation the smartest person and being Helen’s disinterested friend, one of those friends that brilliant women always have, friends of men who can never turn into the role of lovers, Bilibin once in a petit comite [small intimate circle] expressed to his friend Helen his view on this whole matter.
- Ecoutez, Bilibine (Helen always called friends like Bilibine by their last name) - and she touched her white ringed hand to the sleeve of his tailcoat. – Dites moi comme vous diriez a une s?ur, que dois je faire? Lequel des deux? [Listen, Bilibin: tell me, how would you tell your sister, what should I do? Which of the two?]
Bilibin gathered the skin above his eyebrows and thought with a smile on his lips.
“Vous ne me prenez pas en taken aback, vous savez,” he said. - Comme veritable ami j"ai pense et repense a votre affaire. Voyez vous. Si vous epousez le prince (it was a young man)," he bent his finger, "vous perdez pour toujours la chance d"epouser l"autre, et puis vous mecontentez la cour. vous epousant, [You will not take me by surprise, you know. Like a true friend, I have been thinking about your matter for a long time. You see: if you marry a prince, then you will forever lose the opportunity to be the wife of another, and in addition, the court will be dissatisfied. (You know, after all, kinship is involved here.) And if you marry the old count, then you will make happiness last days him, and then... it will no longer be humiliating for the prince to marry the widow of a nobleman.] - and Bilibin loosened his skin.
– Voila un veritable ami! - said the beaming Helen, once again touching Bilibip’s sleeve with her hand. – Mais c"est que j"aime l"un et l"autre, je ne voudrais pas leur faire de chagrin. Je donnerais ma vie pour leur bonheur a tous deux, [Here is a true friend! But I love both of them and I wouldn’t want to upset anyone. For the happiness of both, I would be ready to sacrifice my life.] - she said.
Bilibin shrugged his shoulders, expressing that even he could no longer help such grief.
“Une maitresse femme! Voila ce qui s"appelle poser carrement la question. Elle voudrait epouser tous les trois a la fois", ["Well done woman! That's what is called firmly asking the question. She would like to be the wife of all three at the same time."] - thought Bilibin.

Lesson topic

Isosceles triangle

The purpose of the lesson

Introduce students to an isosceles triangle;
Continue to develop skills in constructing right triangles;
Expand schoolchildren’s knowledge about the properties of isosceles triangles;
Strengthen theoretical knowledge when solving problems.

Lesson Objectives

Be able to formulate, prove and use the theorem about the properties of an isosceles triangle in the process of solving problems;
Continue to develop conscious perception educational material, logical thinking, self-control and self-esteem skills;
Arouse cognitive interest in mathematics lessons;
Foster activity, curiosity and organization.

Lesson Plan

1. General concepts and definitions of an isosceles triangle.
2. Properties of an isosceles triangle.
3. Signs of an isosceles triangle.
4. Questions and tasks.

Isosceles triangle

An isosceles triangle is a triangle that has two equal sides, called the sides of an isosceles triangle, and its third side is called the base.

The top of a given figure is the one located opposite its base.

The angle that lies opposite the base is called the vertex angle of this triangle, and the other two angles are called the base angles of an isosceles triangle.

Types of isosceles triangles

An isosceles triangle, like other figures, can have different types. Among isosceles triangles there are acute, rectangular, obtuse and equilateral triangles.

An acute triangle has all acute angles.
A right triangle has a straight angle at its apex and sharp angles at its base.
An obtuse angle has an obtuse angle at the apex, and the angles at its base are acute.
An equilateral object has all its angles and sides equal.

Properties of an isosceles triangle

Opposite angles in relation to equal sides of an isosceles triangle are equal to each other;

Bisectors, medians and altitudes drawn from angles opposite equal sides of a triangle are equal to each other.

The bisector, median and height, directed and drawn to the base of the triangle, coincide with each other.

The centers of the inscribed and circumscribed circles lie at the altitude, bisector and median (they coincide) drawn to the base.

Angles opposite equal sides of an isosceles triangle are always acute.

These properties of an isosceles triangle are used in solving problems.

Homework

1. Define an isosceles triangle.
2. What is special about this triangle?
3. How does an isosceles triangle differ from a right triangle?
4. Name the properties of an isosceles triangle that you know.
5. Do you think it is possible in practice to check the equality of angles at the base and how to do this?

Exercise

Now let's conduct a short survey and find out how you learned the new material.

Listen carefully to the questions and answer whether the following statement is true:

1. Can a triangle be considered isosceles if its two sides are equal?
2. A bisector is a segment that connects the vertex of a triangle with the middle opposite side?
3. A bisector is a segment that bisects an angle that connects a vertex with a point on the opposite side?

Tips for solving isosceles triangle problems:

1. To determine the perimeter of an isosceles triangle, it is enough to multiply the length of the side by 2 and add this product with the length of the base of the triangle.
2. If the perimeter and length of the base of an isosceles triangle are known in the problem, then to find the length of the side it is enough to subtract the length of the base from the perimeter and divide the found difference by 2.
3. And to find the length of the base of an isosceles triangle, knowing both the perimeter and the length of the side, you just need to multiply the side by two and subtract this product from the perimeter of our triangle.

Tasks:

1. Among the triangles in the figure, identify one extra one and explain your choice:



2. Determine which of the triangles shown in the figure are isosceles, name their bases and sides, and also calculate their perimeter.



3. The perimeter of an isosceles triangle is 21 cm. Find the sides of this triangle if one of them is 3 cm larger. How many solutions can it have? this task?

4. It is known that if side and the angle opposite to the base of one isosceles triangle is equal to the side and angle of the other, then these triangles will be equal. Prove this statement.

5. Think and say whether any isosceles triangle is equilateral? And will any equilateral triangle be isosceles?

6. If the sides of an isosceles triangle are 4 m and 5 m, then what will be its perimeter? How many solutions can this problem have?

7. If one of the angles of an isosceles triangle is equal to 91 degrees, then what are the other angles equal to?

8. Think and answer, what angles should a triangle have in order for it to be both rectangular and isosceles?

How many of you know what Pascal's triangle is? The problem of constructing Pascal's triangle is often asked to test basic programming skills. In general, Pascal's triangle relates to combinatorics and probability theory. So what kind of triangle is this?

Pascal's triangle is an infinite arithmetic triangle or triangle-shaped table that is formed using binomial coefficients. In simple words, the vertex and sides of this triangle are units, and it itself is filled with the sums of the two numbers that are located above. Such a triangle can be folded ad infinitum, but if we outline it, we will get an isosceles triangle with symmetrical lines relative to its vertical axis.



Think about where in Everyday life Have you ever come across isosceles triangles? Isn't it true, the roofs of houses and ancient architectural structures do they look a lot like them? And remember what the basis is Egyptian pyramids? Where else have you come across isosceles triangles?

Since ancient times, isosceles triangles have helped the Greeks and Egyptians in determining distances and heights. For example, the ancient Greeks used it to determine from afar the distance to a ship at sea. And the ancient Egyptians determined the height of their pyramids based on the length of the cast shadow, because... it was an isosceles triangle.

Since ancient times, people already appreciated the beauty and practicality of this figure, since the shapes of triangles surround us everywhere. Moving through different villages, we see the roofs of houses and other buildings that remind us of an isosceles triangle; going into a store, we see packages with triangular-shaped products and juices and even some human faces have the shape of a triangle. This figure is so popular that you can see it at every step.

Subjects > Mathematics > Mathematics 7th grade Definition 7. Any triangle whose two sides are equal is called isosceles.
Two equal sides are called lateral, the third is called the base.
Definition 8. If all three sides of a triangle are equal, then the triangle is called equilateral.
It is a special type of isosceles triangle.
Theorem 18. The height of an isosceles triangle, lowered to the base, is at the same time the bisector of the angle between equal sides, the median and the axis of symmetry of the base.
Proof. Let us lower the height to the base of the isosceles triangle. She will divide it into two equal parts (along the leg and hypotenuse) right triangle. Angles A and C are equal, and the height also divides the base in half and will be the axis of symmetry of the entire figure under consideration.
This theorem can also be formulated as follows:
Theorem 18.1. The median of an isosceles triangle, lowered to the base, is also the bisector of the angle between equal sides, the height and the axis of symmetry of the base.
Theorem 18.2. The bisector of an isosceles triangle, lowered to the base, is simultaneously the height, median and axis of symmetry of the base.
Theorem 18.3. The axis of symmetry of an isosceles triangle is simultaneously the bisector of the angle between equal sides, the median and the altitude.
The proof of these consequences also follows from the equality of the triangles into which an isosceles triangle is divided.

Theorem 19. The angles at the base of an isosceles triangle are equal.
Proof. Let us lower the height to the base of the isosceles triangle. It will divide it into two equal (along the leg and hypotenuse) right triangles, which means the corresponding angles are equal, i.e. ∠ A=∠ C
The criteria for an isosceles triangle come from Theorem 1 and its corollaries and Theorem 2.
Theorem 20. If two of the indicated four lines (height, median, bisector, axis of symmetry) coincide, then the triangle will be isosceles (which means all four lines will coincide).
Theorem 21. If any two angles of a triangle are equal, then it is isosceles.

Proof: Similar to the proof of the direct theorem, but using the second criterion for the equality of triangles. The center of gravity, the centers of the circumcircle and incircle, and the point of intersection of the altitudes of an isosceles triangle all lie on its axis of symmetry, i.e. on high.
An equilateral triangle is isosceles for each pair of its sides. Due to the equality of all its sides, all three angles of such a triangle are equal. Considering that the sum of the angles of any triangle is equal to two right angles, we see that each of the angles of an equilateral triangle is equal to 60°. Conversely, to ensure that all sides of a triangle are equal, it is enough to check that two of its three angles are equal to 60°.
Theorem 22 . In an equilateral triangle, all the remarkable points coincide: the center of gravity, the centers of the inscribed and circumscribed circles, the point of intersection of the altitudes (called the orthocenter of the triangle).
Theorem 23 . If two of the indicated four points coincide, then the triangle will be equilateral and, as a consequence, all four named points will coincide.
Indeed, such a triangle will turn out, according to the previous one, isosceles with respect to any pair of sides, i.e. equilateral. An equilateral triangle is also called a regular triangle. The area of ​​an isosceles triangle is equal to half the product of the square of the side side and the sine of the angle between the sides
Consider this formula for an equilateral triangle, then the alpha angle will be equal to 60 degrees. Then the formula will change to this:

Theorem d1 . In an isosceles triangle, the medians drawn to the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its medians. Then triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have common side AB, sides AL and BK are equal as halves of the lateral sides of an isosceles triangle, and angles LAB and KBA are equal as the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB are equal. But AK and LB are the medians of an isosceles triangle drawn to its lateral sides.
Theorem d2 . In an isosceles triangle, the bisectors drawn to the lateral sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its bisectors. Triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have a common side AB, angles LAB and KBA are equal as the base angles of an isosceles triangle, and angles LBA and KAB are equal as half the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB - the bisectors of triangle ABC - are congruent. The theorem has been proven.
Theorem d3 . In an isosceles triangle, the heights lowered to the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its altitudes. Then angles ABL and KAB are equal, since angles ALB and AKB are right angles, and angles LAB and ABK are equal as the base angles of an isosceles triangle. Consequently, triangles ALB and AKB are equal according to the second criterion for the equality of triangles: they have common side AB, angles KAB and LBA are equal according to the above, and angles LAB and KBA are equal as the base angles of an isosceles triangle. If the triangles are congruent, their sides AK and BL are also congruent. Q.E.D.

The first historians of our civilization - the ancient Greeks - mention Egypt as the birthplace of geometry. It is difficult to disagree with them, knowing with what amazing precision the giant tombs of the pharaohs were erected. Mutual arrangement planes of the pyramids, their proportions, orientation to the cardinal points - to achieve such perfection would be unthinkable without knowing the basics of geometry.

The word “geometry” itself can be translated as “measurement of the earth.” Moreover, the word “earth” does not appear as a planet - part solar system, but as a plane. Marking areas for maintenance Agriculture, most likely, is the very original basis of the science of geometric figures, their types and properties.

A triangle is the simplest spatial figure of planimetry, containing only three points - vertices (there are no fewer). The basis of the foundations, perhaps that is why something mysterious and ancient seems to be in him. All-seeing eye inside a triangle is one of the earliest known occult signs, and the geography of its distribution and time frame are simply amazing. From ancient Egyptian, Sumerian, Aztec and other civilizations to more modern communities of occult lovers scattered across the globe.

What are triangles?

An ordinary scalene triangle is a closed geometric figure, consisting of three segments different lengths and three angles, none of which are right. In addition to this, there are several special types.

An acute triangle has all angles less than 90 degrees. In other words, all the angles of such a triangle are acute.

A right triangle, over which schoolchildren have always cried because of the abundance of theorems, has one angle of 90 degrees or, as it is also called, a straight line.

An obtuse triangle is distinguished by the fact that one of its angles is obtuse, that is, its size is more than 90 degrees.

An equilateral triangle has three sides of equal length. In such a figure, all angles are also equal.

And finally, an isosceles triangle has three sides, two equal to each other.

Distinctive features

The properties of an isosceles triangle also determine its main, main difference - the equality of its two sides. These equal sides are usually called the hips (or, more often, the sides), and the third side is called the “base”.

In the figure under consideration, a = b.

The second criterion for an isosceles triangle follows from the theorem of sines. Since sides a and b are equal, the sines of their opposite angles are equal:

a/sin γ = b/sin α, whence we have: sin γ = sin α.

From the equality of sines follows the equality of angles: γ = α.

So, the second sign of an isosceles triangle is the equality of two angles adjacent to the base.

Third sign. In a triangle, there are such elements as altitude, bisector and median.

If, in the process of solving the problem, it turns out that in the triangle in question any two of these elements coincide: the height with the bisector; bisector with median; median with height - we can definitely conclude that the triangle is isosceles.

Geometric properties of a figure

1. Properties of an isosceles triangle. One of the distinctive qualities of the figure is the equality of the angles adjacent to the base:

<ВАС = <ВСА.

2. One more property was discussed above: the median, bisector and altitude in an isosceles triangle coincide if they are built from its vertex to its base.

3. Equality of bisectors drawn from the vertices at the base:

If AE is the bisector of angle BAC, and CD is the bisector of angle BCA, then: AE = DC.

4. The properties of an isosceles triangle also provide for the equality of the heights that are drawn from the vertices at the base.

If we construct the altitudes of triangle ABC (where AB = BC) from vertices A and C, then the resulting segments CD and AE will be equal.

5. The medians drawn from the corners at the base will also be equal.

So, if AE and DC are medians, that is, AD = DB, and BE = EC, then AE = DC.

Height of an isosceles triangle

The equality of the sides and angles with them introduces some features into the calculation of the lengths of the elements of the figure under consideration.

The altitude in an isosceles triangle divides the figure into 2 symmetrical right triangles, the hypotenuses of which are on the sides. The height in this case is determined according to the Pythagorean theorem as a leg.

A triangle can have all three sides equal, then it will be called equilateral. The height in an equilateral triangle is determined in a similar way, only for calculations it is enough to know only one value - the length of the side of this triangle.

You can determine the height in another way, for example, by knowing the base and the angle adjacent to it.

Median of an isosceles triangle

The type of triangle under consideration, due to its geometric features, can be solved quite simply using a minimal set of initial data. Since the median in an isosceles triangle is equal to both its height and its bisector, the algorithm for determining it is no different from the procedure for calculating these elements.

For example, you can determine the length of the median by the known lateral side and the magnitude of the apex angle.

How to determine the perimeter

Since the two sides of the planimetric figure under consideration are always equal, to determine the perimeter it is enough to know the length of the base and the length of one of the sides.

Let's consider an example when you need to determine the perimeter of a triangle using a known base and height.

The perimeter is equal to the sum of the base and twice the length of the side. The lateral side, in turn, is defined using the Pythagorean theorem as the hypotenuse of a right triangle. Its length is equal to the square root of the sum of the square of the height and the square of half the base.

Area of ​​an isosceles triangle

As a rule, calculating the area of ​​an isosceles triangle does not cause difficulties. The universal rule for determining the area of ​​a triangle as half the product of the base and its height is applicable, of course, in our case. However, the properties of an isosceles triangle again make the task easier.

Let us assume that the height and angle adjacent to the base are known. It is necessary to determine the area of ​​the figure. This can be done this way.

Since the sum of the angles of any triangle is 180°, it is not difficult to determine the size of the angle. Next, using the proportion compiled according to the theorem of sines, the length of the base of the triangle is determined. Everything, base and height - sufficient data to determine the area - are available.

Other properties of an isosceles triangle

The position of the center of a circle circumscribed around an isosceles triangle depends on the magnitude of the vertex angle. So, if an isosceles triangle is acute, the center of the circle is located inside the figure.

The center of a circle circumscribed around an obtuse isosceles triangle lies outside it. And finally, if the angle at the vertex is 90°, the center lies exactly in the middle of the base, and the diameter of the circle passes through the base itself.

In order to determine the radius of a circle circumscribed about an isosceles triangle, it is enough to divide the length of the side by twice the cosine of half the vertex angle.

Among all triangles, there are two special types: right triangles and isosceles triangles. Why are these types of triangles so special? Well, firstly, such triangles extremely often turn out to be the main characters in the problems of the Unified State Exam in the first part. And secondly, problems about right and isosceles triangles are much easier to solve than other geometry problems. You just need to know a few rules and properties. All the most interesting things about right triangles are discussed in, but now let’s look at isosceles triangles. And first of all, what is an isosceles triangle? Or, as mathematicians say, what is the definition of an isosceles triangle?

See what it looks like:

Like a right triangle, an isosceles triangle has special names for its sides. Two equal sides are called sides, and the third party - basis.

And again pay attention to the picture:

It could, of course, be like this:

So be careful: lateral side - one of two equal sides in an isosceles triangle, and the basis is a third party.

Why is an isosceles triangle so good? To understand this, let's draw the height to the base. Do you remember what height is?

What happened? From one isosceles triangle we get two rectangular ones.

This is already good, but this will happen in any, even the most “oblique” triangle.

How is the picture different for an isosceles triangle? Look again:

Well, firstly, of course, it is not enough for these strange mathematicians to just see - they must certainly prove. Otherwise, suddenly these triangles are slightly different, but we will consider them the same.

But don't worry: in this case, proving is almost as easy as seeing.

Shall we start? Look closely, we have:

And that means! Why? Yes, we will simply find and, and from the Pythagorean theorem (remembering at the same time that)

Are you sure? Well, now we have

And on three sides - the easiest (third) sign of equality of triangles.

Well, our isosceles triangle has divided into two identical rectangular ones.

See how interesting it is? It turned out that:

How do mathematicians usually talk about this? Let's go in order:

(Remember here that the median is a line drawn from a vertex that divides the side in half, and the bisector is the angle.)

Well, here we discussed what good things can be seen if given an isosceles triangle. We deduced that in an isosceles triangle the angles at the base are equal, and the height, bisector and median drawn to the base coincide.

And now another question arises: how to recognize an isosceles triangle? That is, as mathematicians say, what are signs of an isosceles triangle?

And it turns out that you just need to “turn” all the statements the other way around. This, of course, does not always happen, but an isosceles triangle is still a great thing! What happens after the “turnover”?

Well, look:
If the height and median coincide, then:


If the height and bisector coincide, then:


If the bisector and the median coincide, then:


Well, don’t forget and use:

  • If you are given an isosceles triangular triangle, feel free to draw the height, get two right triangles and solve the problem about a right triangle.
  • If given that two angles are equal, then a triangle exactly isosceles and you can draw the height and….(The House That Jack Built…).
  • If it turns out that the height is divided in half, then the triangle is isosceles with all the ensuing bonuses.
  • If it turns out that the height divides the angle between the floors - it is also isosceles!
  • If a bisector divides a side in half or a median divides an angle, then this also happens only in an isosceles triangle

Let's see what it looks like in tasks.

Problem 1(the simplest)

In a triangle, sides and are equal, a. Find.

We decide:

First the drawing.

What is the basis here? Certainly, .

Let's remember what if, then and.

Updated drawing:

Let's denote by. What is the sum of the angles of a triangle? ?

We use:

That's answer: .

Not difficult, right? I didn't even have to adjust the height.

Problem 2(Also not very tricky, but we need to repeat the topic)

In a triangle, . Find.

We decide:

The triangle is isosceles! We draw the height (this is the trick with which everything will be decided now).

Now let’s “cross out from life”, let’s just look at it.

So, we have:

Let's remember the table values ​​of cosines (well, or look at the cheat sheet...)

All that remains is to find: .

Answer: .

Note that we here Very required knowledge regarding right triangles and “tabular” sines and cosines. Very often this happens: the topics , “Isosceles triangle” and in problems go together, but are not very friendly with other topics.

Isosceles triangle. Average level.

These two equal sides are called sides, A the third side is the base of an isosceles triangle.

Look at the picture: and - the sides, - the base of the isosceles triangle.

Let's use one picture to understand why this happens. Let's draw a height from a point.

This means that all corresponding elements are equal.

All! In one fell swoop (height) they proved all the statements at once.

And remember: to solve a problem about an isosceles triangle, it is often very useful to lower the height to the base of the isosceles triangle and divide it into two equal right triangles.

Signs of an isosceles triangle

The converse statements are also true:

Almost all of these statements can again be proven “in one fell swoop.”

1. So, let in turned out to be equal and.

Let's check the height. Then

2. a) Now let in some triangle height and bisector coincide.

2. b) And if the height and median coincide? Everything is almost the same, no more complicated!

- on two sides

2. c) But if there is no height, which is lowered to the base of an isosceles triangle, then there are no initially right triangles. Badly!

But there is a way out - read it in the next level of the theory, since the proof here is more complicated, but for now just remember that if the median and bisector coincide, then the triangle will also turn out to be isosceles, and the height will still coincide with these bisector and median.

Let's summarize:

  1. If the triangle is isosceles, then the angles at the base are equal, and the altitude, bisector and median drawn to the base coincide.
  2. If in some triangle there are two equal angles, or some two of the three lines (bisector, median, altitude) coincide, then such a triangle is isosceles.

Isosceles triangle. Brief description and basic formulas

An isosceles triangle is a triangle that has two equal sides.

Signs of an isosceles triangle:

  1. If in a certain triangle two angles are equal, then it is isosceles.
  2. If in some triangle they coincide:
    A) height and bisector or
    b) height and median or
    V) median and bisector,
    drawn to one side, then such a triangle is isosceles.

Well, the topic is over. If you are reading these lines, it means you are very cool.

Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!

Now the most important thing.

You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough...

For what?

For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.

I won’t convince you of anything, I’ll just say one thing...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.

You won't be asked for theory during the exam.

You will need solve problems against time.

And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.

It's like in sports - you need to repeat it many times to win for sure.

Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!

You can use our tasks (optional) and we, of course, recommend them.

In order to get better at using our tasks, you need to help extend the life of the YouClever textbook you are currently reading.

How? There are two options:

  1. Unlock all hidden tasks in this article - 299 rub.
  2. Unlock access to all hidden tasks in all 99 articles of the textbook - 999 rub.

Yes, we have 99 such articles in our textbook and access to all tasks and all hidden texts in them can be opened immediately.

In the second case we will give you simulator “6000 problems with solutions and answers, for each topic, at all levels of complexity.” It will definitely be enough to get your hands on solving problems on any topic.

In fact, this is much more than just a simulator - a whole training program. If necessary, you can also use it for FREE.

Access to all texts and programs is provided for the ENTIRE period of the site’s existence.

In conclusion...

If you don't like our tasks, find others. Just don't stop at theory.

“Understood” and “I can solve” are completely different skills. You need both.

Find problems and solve them!



What else to read

Abstract: History of the development of hygiene 1. The development of hygiene in primitive and slave societies. The emergence and development of hygiene has a centuries-old history...
Formation of estimated liabilities and reserves for vacations As in 1s 8 Hello, dear blog readers. Lately, in my consulting work, I have quite often...
Puff khinkal in a pressure cooker Puff DAGESTAN KHINKAL Puff khinkal is often prepared in Dagestan. A very tasty, but high-calorie, masculine dish....