Right triangle- a triangle, one angle of which is right (equal to 90 0). Therefore, the other two angles add up to 90 0.

Sides of a right triangle

The side that is opposite the ninety degree angle is called the hypotenuse. The other two sides are called legs. The hypotenuse is always longer than the legs, but shorter than their sum.

Right triangle. Properties of a triangle

If the leg is opposite an angle of thirty degrees, then its length corresponds to half the length of the hypotenuse. It follows that the angle opposite the leg, the length of which corresponds to half the hypotenuse, is equal to thirty degrees. The leg is equal to the average of the proportional hypotenuse and the projection that the leg gives to the hypotenuse.

Pythagorean theorem

Any right triangle obeys the Pythagorean theorem. This theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse. If we assume that the legs are equal to a and b, and the hypotenuse is c, then we write: a 2 + b 2 = c 2. The Pythagorean theorem is used to solve all geometric problems involving right triangles. It will also help to draw a right angle in the absence of the necessary tools.

Height and median

A right triangle is characterized by the fact that its two altitudes are aligned with its legs. To find the third side, you need to find the sum of the projections of the legs onto the hypotenuse and divide by two. If from the top right angle draw the median, then it will turn out to be the radius of the circle that is described around the triangle. The center of this circle will be the middle of the hypotenuse.

Right triangle. Area and its calculation

The area of ​​right triangles is calculated using any formula for finding the area of ​​a triangle. In addition, you can use another formula: S = a * b / 2, which states that to find the area you need to divide the product of the lengths of the legs by two.

Cosine, sine and tangent right triangle

The cosine of an acute angle is the ratio of the leg adjacent to the angle to the hypotenuse. It is always less than one. Sine is the ratio of the leg that lies opposite the angle to the hypotenuse. Tangent is the ratio of the leg opposite the angle to the leg adjacent to this angle. Cotangent is the ratio of the side adjacent to the angle to the side opposite the angle. Cosine, sine, tangent and cotangent are not dependent on the size of the triangle. Their value is affected only by the degree measure of the angle.

Triangle solution

To calculate the value of the leg opposite the angle, you need to multiply the length of the hypotenuse by the sine of this angle or the size of the second leg by the tangent of the angle. To find the leg adjacent to an angle, it is necessary to calculate the product of the hypotenuse and the cosine of the angle.

Isosceles right triangle

If a triangle has a right angle and equal sides, then it is called an isosceles right triangle. The acute angles of such a triangle are also equal - 45 0 each. The median, bisector and altitude drawn from the right angle of an isosceles right triangle are the same.

Properties of a right triangle

Dear seventh graders, you already know what geometric figures are called triangles, you know how to prove signs of their equality. You also know about special cases of triangles: isosceles and right angles. You are well aware of the properties of isosceles triangles.

But right triangles also have many properties. One obvious thing is related to the sum theorem internal corners Triangle: In a right triangle, the sum of the acute angles is 90°. The most amazing property you will learn about a right triangle in 8th grade when you study the famous Pythagorean theorem.

Now we will talk about two more important properties. One is for 30° right triangles and the other is for random right triangles. Let us formulate and prove these properties.

You are well aware that in geometry it is customary to formulate statements that are converse to proven ones, when the condition and conclusion in the statement change places. Converse statements are not always true. In our case, both converse statements are true.

Property 1.1 In a right triangle, the leg opposite the 30° angle is equal to half the hypotenuse.

Proof: Consider the rectangular ∆ ABC, in which ÐA=90°, ÐB=30°, then ÐC=60°..gif" width="167" height="41">, therefore, what needed to be proved.

Property 1.2 (reverse to property 1.1) If in a right triangle the leg is equal to half the hypotenuse, then the angle opposite it is 30°.

Property 2.1 In a right triangle, the median drawn to the hypotenuse is equal to half the hypotenuse.

Let's consider a rectangular ∆ ABC, in which РВ=90°.

BD-median, that is, AD=DC. Let's prove that .

To prove this, we will make an additional construction: we will continue BD beyond point D so that BD=DN and connect N with A and C..gif" width="616" height="372 src=">

Given: ∆ABC, ÐC=90o, ÐA=30o, ÐBEC=60o, EC=7cm

1. ÐEBC=30o, since in a rectangular ∆BCE the sum of acute angles is 90o

2. BE=14cm(property 1)

3. ÐABE=30o, since ÐA+ÐABE=ÐBEC (property of the external angle of a triangle) therefore ∆AEB is isosceles AE=EB=14cm.

BC=2AN=20 cm (property 2).

Task 3. Prove that the altitude and median of a right triangle taken to the hypotenuse form an angle equal to the difference between the acute angles of the triangle.

Given: ∆ ABC, ÐBAC=90°, AM-median, AH-height.

Prove: RMAN=RS-RV.

Proof:

1)РМАС=РС (by property 2 ∆ AMC-isosceles, AM=SM)

2) ÐMAN = ÐMAS-ÐNAS = ÐS-ÐNAS.

It remains to prove that РНАС=РВ. This follows from the fact that ÐB+ÐC=90° (in ∆ ABC) and ÐNAS+ÐC=90° (from ∆ ANS).

So, RMAN = RС-РВ, which is what needed to be proved.

https://pandia.ru/text/80/358/images/image014_39.gif" width="194" height="184">Given: ∆ABC, ÐBAC=90°, AN-height, .

Find: РВ, РС.

Solution: Let's take the median AM. Let AN=x, then BC=4x and

VM=MS=AM=2x.

In a rectangular ∆AMN, the hypotenuse AM is 2 times larger than the leg AN, therefore ÐAMN=30°. Since VM=AM,

РВ=РВAM100%">

Let's extend AC beyond point A so that AD=AC. Then ∆ABC=∆ABD (on 2 legs). BD=BC=2AC=CD, thus ∆DBC-equilateral, ÐC=60o and ÐABC=30o.

Problem 5

In an isosceles triangle, one of the angles is 120°, the base is 10 cm. Find the height drawn to the side.

Solution: to begin with, we note that the angle of 120° can only be at the vertex of the triangle and that the height drawn to the side will fall on its continuation.

AB - staircase, K - kitten.

In any position of the ladder, until it finally falls to the ground, ∆ABC is rectangular. MC - median ∆ABC.

According to property 2 SK = 1/2AB. That is, at any moment in time the length of the segment SK is constant.

Answer: point K will move along a circular arc with center C and radius SC=1/2AB.

Problems for independent solution.

One of the angles of a right triangle is 60°, and the difference between the hypotenuse and the shorter leg is 4 cm. find the length of the hypotenuse. In a rectangular ∆ ABC with hypotenuse BC and angle B equal to 60°, the height AD is drawn. Find DC if DB=2cm. B ∆ABC ÐC=90o, CD - height, BC=2ВD. Prove that AD=3ВD. The altitude of a right triangle divides the hypotenuse into parts 3 cm and 9 cm. Find the angles of the triangle and the distance from the middle of the hypotenuse to the longer leg. The bisector splits the triangle into two isosceles triangles. Find the angles of the original triangle. The median splits the triangle into two isosceles triangles. Is it possible to find angles

The original triangle?

Average level

Right triangle. The Complete Illustrated Guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,

and in this

and in this

What's good about a right triangle? Well..., firstly, there are special beautiful names for its sides.

Attention to the drawing!

Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!

Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. Pythagoras proved it completely time immemorial, and since then she has brought a lot of benefit to those who know her. And the best thing about it is that it is simple.

So, Pythagorean theorem:

Do you remember the joke: “Pythagorean pants are equal on all sides!”?

Let's draw these same Pythagorean pants and look at them.

Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."

Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.

In this picture, the sum of the areas of the small squares is equal to the area of ​​the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem?

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:

It should be easy now:

Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's move on... to dark forest... trigonometry! To the terrible words sine, cosine, tangent and cotangent.

Sine, cosine, tangent, cotangent in a right triangle.

In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!

1.
Actually it sounds like this:

What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course have! This is a leg!

What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and

Now, pay attention! Look what we got:

See how cool it is:

Now let's move on to tangent and cotangent.

How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?

See how the numerator and denominator have swapped places?

And now the corners again and made an exchange:

Summary

Let's briefly write down everything we've learned.

Pythagorean theorem:

The main theorem about right triangles is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge

It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

See how cleverly we divided its sides into lengths and!

Now let's connect the marked dots

Here we, however, noted something else, but you yourself look at the drawing and think why this is so.

What is the area of ​​the larger square?

Right, .

What about a smaller area?

Certainly, .

The total area of ​​the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses.

What happened? Two rectangles. This means that the area of ​​the “cuts” is equal.

Let's put it all together now.

Let's convert:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite side to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.

The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.

And once again all this in the form of a tablet:

It is very comfortable!

Signs of equality of right triangles

I. On two sides

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:

THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both it was opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles?

Take a look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides.

But for the equality of right triangles, only two corresponding elements are enough. Great, right?

The situation is approximately the same with the signs of similarity of right triangles.

Signs of similarity of right triangles

I. Along an acute angle

II. On two sides

III. By leg and hypotenuse

Median in a right triangle

Why is this so?

Instead of a right triangle, consider a whole rectangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it turned out that

1. - median:

Remember this fact! Helps a lot!

What’s even more surprising is that the opposite is also true.

What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look carefully. We have: , that is, the distances from the point to all three peaks the triangles turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?

So let's start with this “besides...”.

Let's look at and.

But similar triangles have all equal angles!

The same can be said about and

Now let's draw it together:

What benefit can be derived from this “triple” similarity?

Well, for example - two formulas for the height of a right triangle.

Let us write down the relations of the corresponding parties:

To find the height, we solve the proportion and get the first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

You need to remember both of these formulas very well and use the one that is more convenient.

Let's write them down again

Pythagorean theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .

Signs of equality of right triangles:

• on two sides:
• by leg and hypotenuse: or
• along the leg and adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one acute corner: or
• from the proportionality of two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

• via legs:
• through the leg and sharp corner: .

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 successful passing the Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.

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

People who received a good education, earn much more than those who did not receive 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!

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

Find problems and solve them!

The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is located opposite the angle of 90 degrees. Pythagorean triangle is called the one whose sides are equal natural numbers; their lengths in this case are called “Pythagorean triple”.

Egyptian triangle

In order to current generation learned geometry in the form in which it is taught in school now, it has developed over several centuries. The fundamental point is considered to be the Pythagorean theorem. The sides of a rectangular is known throughout the world) are 3, 4, 5.

Few people are not familiar with the phrase “Pythagorean pants are equal in all directions.” However, in reality the theorem sounds like this: c 2 (square of the hypotenuse) = a 2 + b 2 (sum of squares of the legs).

Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called “Egyptian”. The interesting thing is that which is inscribed in the figure is equal to one. The name arose around the 5th century BC, when Greek philosophers traveled to Egypt.

When building the pyramids, architects and surveyors used the ratio 3:4:5. Such structures turned out to be proportional, pleasant to look at and spacious, and also rarely collapsed.

In order to build a right angle, the builders used a rope with 12 knots tied on it. In this case, the probability of constructing a right triangle increased to 95%.

Signs of equality of figures

• An acute angle in a right triangle and a long side, which are equal to the same elements in the second triangle, are an indisputable sign of equality of figures. Taking into account the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are identical according to the second criterion.
• When superimposing two figures on top of each other, rotate them so that, when combined, they become one isosceles triangle. According to its property, the sides, or rather the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.

Based on the first sign, it is very easy to prove that the triangles are indeed equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.

The triangles will be identical according to the second criterion, the essence of which is the equality of the leg and the acute angle.

Properties of a triangle with a right angle

The height that is lowered from the right angle splits the figure into two equal parts.

The sides of a right triangle and its median can be easily recognized by the rule: the median that falls on the hypotenuse is equal to half of it. can be found both by Heron's formula and by the statement that it is equal to half the product of the legs.

In a right triangle, the properties of angles of 30°, 45° and 60° apply.

• With an angle of 30°, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
• If the angle is 45°, then the second acute angle is also 45°. This suggests that the triangle is isosceles and its legs are the same.
• The property of an angle of 60° is that the third angle has a degree measure of 30°.

The area can be easily found out using one of three formulas:

1. through the height and the side on which it descends;
2. according to Heron's formula;
3. on the sides and the angle between them.

The sides of a right triangle, or rather the legs, converge with two altitudes. In order to find the third, it is necessary to consider the resulting triangle, and then, using the Pythagorean theorem, calculate the required length. In addition to this formula, there is also a relationship between twice the area and the length of the hypotenuse. The most common expression among students is the first one, as it requires fewer calculations.

Theorems applying to right triangle

Right triangle geometry involves the use of theorems such as:

Average level

Right triangle. The Complete Illustrated Guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,

and in this

and in this

What's good about a right triangle? Well..., firstly, there are special beautiful names for its sides.

Attention to the drawing!

Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!

Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. And the best thing about it is that it is simple.

So, Pythagorean theorem:

Do you remember the joke: “Pythagorean pants are equal on all sides!”?

Let's draw these same Pythagorean pants and look at them.

Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."

Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.

In this picture, the sum of the areas of the small squares is equal to the area of ​​the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem?

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:

It should be easy now:

Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's go further... into the dark forest... of trigonometry! To the terrible words sine, cosine, tangent and cotangent.

Sine, cosine, tangent, cotangent in a right triangle.

In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!

1.
Actually it sounds like this:

What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course have! This is a leg!

What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and

Now, pay attention! Look what we got:

See how cool it is:

Now let's move on to tangent and cotangent.

How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?

See how the numerator and denominator have swapped places?

And now the corners again and made an exchange:

Summary

Let's briefly write down everything we've learned.

Pythagorean theorem:

The main theorem about right triangles is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge

It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

See how cleverly we divided its sides into lengths and!

Now let's connect the marked dots

Here we, however, noted something else, but you yourself look at the drawing and think why this is so.

What is the area of ​​the larger square?

Right, .

What about a smaller area?

Certainly, .

The total area of ​​the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses.

What happened? Two rectangles. This means that the area of ​​the “cuts” is equal.

Let's put it all together now.

Let's convert:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite side to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.

The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.

And once again all this in the form of a tablet:

It is very comfortable!

Signs of equality of right triangles

I. On two sides

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:

THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both it was opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles?

Take a look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides.

But for the equality of right triangles, only two corresponding elements are enough. Great, right?

The situation is approximately the same with the signs of similarity of right triangles.

Signs of similarity of right triangles

I. Along an acute angle

II. On two sides

III. By leg and hypotenuse

Median in a right triangle

Why is this so?

Instead of a right triangle, consider a whole rectangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it turned out that

1. - median:

Remember this fact! Helps a lot!

What’s even more surprising is that the opposite is also true.

What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?

So let's start with this “besides...”.

Let's look at and.

But similar triangles have all equal angles!

The same can be said about and

Now let's draw it together:

What benefit can be derived from this “triple” similarity?

Well, for example - two formulas for the height of a right triangle.

Let us write down the relations of the corresponding parties:

To find the height, we solve the proportion and get the first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

You need to remember both of these formulas very well and use the one that is more convenient.

Let's write them down again

Pythagorean theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .

Signs of equality of right triangles:

• on two sides:
• by leg and hypotenuse: or
• along the leg and adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one acute corner: or
• from the proportionality of two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

• via legs:
• through a leg and an acute angle: .

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!

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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!