It doesn’t matter which school curriculum contains such a subject as geometry. Each of us, as a student, studied this discipline and solved certain problems. But for many people, their school years are behind them and some of the acquired knowledge has been erased from memory.

But what if you suddenly need to find the answer to a certain question from a school textbook, for example, how to find the height in a right triangle? IN in this case A modern advanced computer user will first open the Internet and find the information that interests him.

Basic information about triangles

This geometric figure consists of 3 segments connected to each other at the end points, and the points of contact of these points are not on the same straight line. The segments that make up a triangle are called its sides. The junctions of the sides form the tops of the figure, as well as its corners.

Types of triangles depending on angles

This figure can have 3 types of angles: sharp, obtuse and straight. Depending on this, among the triangles the following varieties are distinguished:

Types of triangles depending on the length of the sides

As mentioned earlier, this figure appears from 3 segments. Based on their size, the following types of triangles are distinguished:

How to find the height of a right triangle

Two similar sides of a right triangle that form a right angle at the point of contact are called legs. The segment that connects them is called “hypotenuse”. To find the height in a given geometric figure, you need to lower a line from the top right angle to the hypotenuse. With all this, this line should divide the angle of 90? exactly in half. Such a segment is called a bisector.

The picture above shows a right triangle, the height of which we will have to calculate. This can be done in several ways:

If you draw a circle around a triangle and draw a radius, its value will be half the size of the hypotenuse. Based on this, the height of a right triangle can be calculated using the formula:

Any school program includes such a subject as geometry. Each of us, as a student, studied this discipline and solved certain problems. But for many people, their school years are behind them and some of the acquired knowledge has been erased from memory.

But what if you suddenly need to find the answer to some question from a school textbook, for example, how to find the height in a right triangle? In this case, the modern advanced computer user will first open the Internet and find the information that interests him.

Basic information about triangles

This geometric figure consists of 3 segments connected to each other at the end points, and the points of contact of these points are not on the same straight line. The segments that make up a triangle are called its sides. The junctions of the sides form the vertices of the figure, as well as its corners.

Types of triangles depending on angles

This figure can have three types of angles: acute, obtuse and straight. Depending on this, the following types of triangles are distinguished:

Types of triangles depending on the length of the sides

As mentioned earlier, this figure is formed from three segments. Based on their size, the following types of triangles are distinguished:

How to find the height of a right triangle

Two identical sides of a right triangle that form a right angle at the point of contact are called legs. The segment that connects them is called the “hypotenuse”. To find the height in a given geometric figure, you need to lower a line from the vertex of the right angle to the hypotenuse. In this case, this line should divide the 90º angle exactly in half. Such a segment is called a bisector.

The picture above shows right triangle, height which we will have to calculate. This can be done in several ways:

If you draw a circle around a triangle and draw a radius, its value will be half the size of the hypotenuse. Based on this, the height of a right triangle can be calculated using the formula:

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Right triangle - this is a triangle in which one of the angles is straight, that is, equal to 90 degrees.

• The side opposite the right angle is called the hypotenuse (in the figure indicated as c or AB)
• The side adjacent to the right angle is called the leg. Each right triangle has two legs (in the figure they are designated as a and b or AC and BC)

Formulas and properties of a right triangle

(see picture above)

a, b- legs of a right triangle

c- hypotenuse

α, β - acute angles of a triangle

S- square

h- height lowered from the vertex of a right angle to the hypotenuse

m a a from the opposite corner ( α )

m b- median drawn to the side b from the opposite corner ( β )

m c- median drawn to the side c from the opposite corner ( γ )

IN right triangle any of the legs is less than the hypotenuse(Formula 1 and 2). This property is a consequence of the Pythagorean theorem.

Cosine of any of the acute angles less than one (Formula 3 and 4). This property follows from the previous one. Since any of the legs is less than the hypotenuse, the ratio of leg to hypotenuse is always less than one.

The square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean theorem). (Formula 5). This property is constantly used when solving problems.

Area of ​​a right triangle equal to half the product of legs (Formula 6)

Sum of squared medians to the legs is equal to five squares of the median to the hypotenuse and five squares of the hypotenuse divided by four (Formula 7). In addition to the above, there is 5 more formulas, therefore, it is recommended that you also read the lesson “Median of a Right Triangle,” which describes the properties of the median in more detail.

Height of a right triangle is equal to the product of the legs divided by the hypotenuse (Formula 8)

The squares of the legs are inversely proportional to the square of the height lowered to the hypotenuse (Formula 9). This identity is also one of the consequences of the Pythagorean theorem.

Hypotenuse length equal to the diameter (two radii) of the circumscribed circle (Formula 10). Hypotenuse of a right triangle is the diameter of the circumcircle. This property is often used in problem solving.

Inscribed radius V right triangle circle can be found as half of the expression including the sum of the legs of this triangle minus the length of the hypotenuse. Or as the product of legs divided by the sum of all sides (perimeter) of a given triangle. (Formula 11)
Sine of angle relation to the opposite this angle leg to hypotenuse(by definition of sine). (Formula 12). This property is used when solving problems. Knowing the sizes of the sides, you can find the angle they form.

The cosine of angle A (α, alpha) in a right triangle will be equal to attitude adjacent this angle leg to hypotenuse(by definition of sine). (Formula 13)

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... first of all, there are special beautiful names for his 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 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 on 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.

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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!

Triangle - This is one of the most famous geometric figures. It is used everywhere - not only in drawings, but also as interior items, parts various designs and buildings. There are several types of this figure - rectangular is one of them. His distinctive feature is the presence of a right angle equal to 90°. To find two of the three heights, it is enough to measure the legs. The third is the value between the vertex of the right angle and the middle of the hypotenuse. Often in geometry the question is how to find the height of a right triangle. Let's solve this simple problem.

Necessary:

- ruler;
– a book on geometry;
- right triangle.

Instructions:

• Draw a triangle with a right angle ABC, where is the angle ABC equals 90 ° , that is, it is direct. Lower the height H from a right angle to the hypotenuse - a segment AS. Mark the place where the segments touch with a dot. D.
• You should now have another triangle - A.D.B.. Please note that it is similar to the existing one ABC, since the angles ABS And ADB = 90°, then they are equal to each other, and the angle BAD is common to both geometric figures. By comparing them, we can conclude that the parties AD/AB = BD/BS = AB/AS. From the resulting relations it can be concluded that AD equals AB²/AS.
• Since the resulting triangle A.D.B. has a right angle, when measuring its sides and hypotenuse, you can use the Pythagorean theorem. Here's what it looks like: AB² = AD² + BD². To solve it, use the resulting equality AD. You should get the following: BD² = AB² - (AB²/AC)². Since the triangle being measured ABS is rectangular, then BS² equals AS² — AB². Therefore, the side BD² equals AB²BC²/AC², which with extraction of the root will be equal to BD = AB*BS/AS.
• Similarly, the solution can be derived using another resulting triangle -
  BDS. In this case, it is also similar to the original ABC, thanks to two angles - ABS And BDS = 90°, and the angle DSB is common. Further, as in the previous example, the proportion is displayed in the aspect ratio, where BD/AB = DS/BS = BS/AS. Hence the value D.S. is derived through equality BS²/AS. Because, AB² = AD*AS , That BS² = DS*AS. From this we conclude that BD² = (AB*BS/AS)² or AD*AS*DS*AS/AS², which equals AD*DS. To find the height in this case, it is enough to remove the root from the product D.S. And AD.