Definition.Right triangle - triangle, one of the angles of which is right (equal).

A right triangle is a special case of an ordinary triangle. Therefore, all the properties of ordinary triangles for rectangular ones are preserved. But there are some particular properties due to the presence of a right angle.

Common notation (Fig. 1):

- right angle;

- hypotenuse;

- legs;

.

Rice. one.

Withright triangle properties.

Property 1. The sum of the angles and a right triangle is .

Proof. Recall that the sum of the angles of any triangle is . Considering the fact that , we get that the sum of the remaining two angles is That is,

Property 2. In a right triangle hypotenuse more than any of legs(is the largest side).

Proof. Recall that in a triangle opposite the larger angle lies the larger side (and vice versa). It follows from property 1 proved above that the sum of the angles and the right triangle is equal to . Since the angle of a triangle cannot be 0, each of them is less than . This means that it is the largest, which means that the largest side of the triangle lies opposite it. Hence, the hypotenuse is the largest side of a right triangle, that is:.

Property 3. In a right triangle, the hypotenuse is less than the sum of the legs.

Proof. This property becomes clear if we recall triangle inequality.

triangle inequality

In any triangle, the sum of any two sides is greater than the third side.

Property 3 immediately follows from this inequality.

Note: despite the fact that each of the legs individually is less than the hypotenuse, their sum turns out to be greater. In a numerical example, it looks like this: , but .

in:

1st sign (on 2 sides and the angle between them): if two triangles have equal sides and the angle between them, then such triangles are congruent.

2nd sign (on the side and two adjacent angles): if triangles have equal side and two angles adjacent to a given side, then such triangles are congruent. Note: using the fact that the sum of the angles of a triangle is constant and equal to , it is easy to prove that the condition of "adjacency" of the angles is not necessary, that is, the sign will be true in the following formulation: "... a side and two angles are equal, then ...".

3rd sign (on 3 sides): if all three sides of a triangle are equal, then such triangles are congruent.

Naturally, all these signs remain true for right triangles. However, right triangles have one essential feature - they always have a pair of equal right angles. Therefore, these signs are simplified for them. So, let's formulate the signs of equality of right triangles:

1st sign (on two legs): if the legs of right triangles are equal in pairs, then such triangles are equal to each other (Fig. 2).

Given:

Rice. 2. Illustration of the first sign of equality of right triangles

Prove:

Proof: in right triangles: . So, we can use the first sign of equality of triangles (on 2 sides and the angle between them) and get: .

2-th sign (on the leg and angle): if the leg and acute angle of one right triangle are equal to the leg and acute angle of another right triangle, then such triangles are equal to each other (Fig. 3).

Given:

Rice. 3. Illustration of the second sign of equality of right triangles

Prove:

Proof: we note right away that the fact that the angles adjacent to equal legs are equal is not fundamental. Indeed, the sum of acute angles of a right triangle (by property 1) is equal to . Hence, if one pair of these angles is equal, then the other is equal (since their sums are the same).

The proof of this feature comes down to using second sign of equality of triangles(at 2 corners and side). Indeed, by condition, the legs and a pair of angles adjacent to them are equal. But the second pair of angles adjacent to them consists of the angles . So, we can use the second criterion for the equality of triangles and get: .

3rd sign (by hypotenuse and angle): if the hypotenuse and the acute angle of one right triangle are equal to the hypotenuse and the acute angle of another right triangle, then such triangles are equal to each other (Fig. 4).

Given:

Rice. 4. Illustration of the third sign of equality of right triangles

Prove:

Proof: to prove this sign, you can immediately use the second sign of the equality of triangles- by the side and two angles (more precisely, by the consequence, which states that the angles do not have to be adjacent to the side). Indeed, by the condition: , , and from the properties of right triangles it follows that . So, we can use the second criterion for the equality of triangles, and get: .

4th sign (by hypotenuse and leg): if the hypotenuse and leg of one right triangle are equal respectively to the hypotenuse and leg of another right triangle, then such triangles are equal to each other (Fig. 5).

Given:

Rice. 5. Illustration of the fourth sign of equality of right triangles

Prove:

Proof: To prove this sign, we will use the sign of equality of triangles, which we formulated and proved in the last lesson, namely: if triangles have equal two sides and a larger angle, then such triangles are equal. Indeed, by condition we have two equal sides. In addition, by the property of right triangles: . It remains to prove that the right angle is the largest in the triangle. Let's assume that this is not the case, which means that there must be at least one more angle that is greater than . But then the sum of the angles of the triangle will already be greater. But this is impossible, which means that such an angle cannot exist in a triangle. Hence, the right angle is the largest in a right triangle. So, you can use the sign formulated above, and get: .

We now formulate one more property, which is characteristic only for right triangles.

Property

The leg opposite the angle at is 2 times smaller than the hypotenuse(Fig. 6).

Given:

Rice. 6.

Prove:AB

Proof: perform an additional construction: extend the line beyond the point by a segment equal to . Let's get a point. Since the angles and are adjacent, their sum is equal to . Since , then the angle .

So right triangles (by two legs: - general, - by construction) - the first sign of the equality of right triangles.

From the equality of triangles follows the equality of all corresponding elements. Means, . Where: . In addition, (from the equality of all the same triangles). This means that the triangle is isosceles (since it has equal angles at the base), but an isosceles triangle, one of whose angles is equal, is equilateral. It follows from this, in particular, that .

Property of the leg opposite the angle in

It is worth noting that the converse statement is also true: if in a right triangle the hypotenuse is twice as large as one of the legs, then the acute angle opposite this leg is equal to.

Note: sign means that if some statement is true, then the triangle is a right triangle. That is, the feature allows you to identify a right triangle.

It is important not to confuse the sign with property- that is, if the triangle is right-angled, then it has such properties ... Often the signs and properties are mutually inverse, but not always. For example, the property of an equilateral triangle: an equilateral triangle has an angle. But this will not be a sign of an equilateral triangle, since not every triangle that has an angle, is equilateral.

Solving geometric problems requires a huge amount of knowledge. One of the fundamental definitions of this science is a right triangle.

This concept means consisting of three corners and

sides, and the value of one of the angles is 90 degrees. The sides that make up a right angle are called the legs, while the third side that is opposite it is called the hypotenuse.

If the legs in such a figure are equal, it is called an isosceles right triangle. In this case, there is an affiliation to two, which means that the properties of both groups are observed. Recall that the angles at the base of an isosceles triangle are absolutely always equal, therefore, the acute angles of such a figure will include 45 degrees each.

The presence of one of the following properties allows us to assert that one right triangle is equal to another:

1. the legs of two triangles are equal;
2. the figures have the same hypotenuse and one of the legs;
3. the hypotenuse and any of the acute angles are equal;
4. the condition of equality of the leg and the acute angle is observed.

The area of ​​a right triangle can be easily calculated both using standard formulas and as a value equal to half the product of its legs.

In a right triangle, the following relations are observed:

1. the leg is nothing but the mean proportional to the hypotenuse and its projection on it;
2. if you describe a circle around a right triangle, its center will be in the middle of the hypotenuse;
3. the height drawn from the right angle is the mean proportional to the projections of the legs of the triangle onto its hypotenuse.

It is interesting that no matter what the right triangle is, these properties are always observed.

Pythagorean theorem

In addition to the above properties, right triangles are characterized by the following condition:

This theorem is named after its founder - the Pythagorean theorem. He discovered this relation when he was studying the properties of squares built on

To prove the theorem, we construct a triangle ABC, whose legs we denote a and b, and the hypotenuse c. Next, we will build two squares. One side will be the hypotenuse, the other the sum of two legs.

Then the area of ​​the first square can be found in two ways: as the sum of the areas of the four triangles ABC and the second square, or as the square of the side, naturally, these ratios will be equal. I.e:

with 2 + 4 (ab/2) = (a + b) 2 , we transform the resulting expression:

c 2 +2 ab = a 2 + b 2 + 2 ab

As a result, we get: c 2 \u003d a 2 + b 2

Thus, the geometric figure of a right triangle corresponds not only to all the properties characteristic of triangles. The presence of a right angle leads to the fact that the figure has other unique relationships. Their study is useful not only in science, but also in everyday life, since such a figure as a right triangle is found everywhere.

Middle level

Right triangle. Complete illustrated guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

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

and in such

and in such

What is good about a right triangle? Well... first of all, there are special beautiful names for his parties.

Attention to the drawing!

Remember and do not confuse: legs - two, and the hypotenuse - only one(the only, unique and longest)!

Well, we 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 many benefits to those who know it. And the best thing about her is that she is simple.

So, Pythagorean theorem:

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

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

Does it really look like 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, more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

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

Doesn't it sound a little different, doesn't it? And so, when Pythagoras drew the statement of his theorem, just such a picture turned 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 the children better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty invented 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 memorize everything with words??! And we can be glad that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to better remember:

Now it should be easy:

The square of the hypotenuse is equal to the sum of the squares of the legs.

Well, the most important theorem about a right triangle was discussed. If you are interested in how it is proved, read the next levels of theory, and now let's move on ... 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 you really don't want to, do you? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is it all 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.
It actually sounds like this:

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

But what about the angle? Look closely. Which leg is adjacent to the corner? Of course, the cat. So, for the angle, the leg is adjacent, and

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

How to put it into words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the cathet? Adjacent to the corner. So what did we get?

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

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

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

You see how cunningly we divided its sides into segments of lengths and!

Now let's connect the marked points

Here we, however, noted something else, but you yourself look at the picture and think about why.

What is the area of ​​the larger square?

Correctly, .

What about the smaller area?

Certainly, .

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

What happened? Two rectangles. So, the area of ​​"cuttings" is equal.

Let's put it all together now.

Let's transform:

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 leg 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 leg to the adjacent leg.

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

And once again, all this in the form of a plate:

It is very comfortable!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! Here it is very important that the legs are "corresponding". For example, if it goes like this:

THEN THE 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 - opposite.

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

Look at the topic “and pay attention to the fact that for the equality of “ordinary” triangles, you need the equality of their three elements: two sides and an angle between them, two angles and a side between them, or three sides.

But for the equality of right-angled triangles, only two corresponding elements are enough. It's great, right?

Approximately the same situation with signs of similarity of right triangles.

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

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 happened that

1. - median:

Remember this fact! Helps a lot!

What is even more surprising is that the converse is also true.

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

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

So let's start with this "besides...".

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What use can be drawn from this "triple" similarity.

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

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get 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:

Both of these formulas must be remembered very well and the one that is more convenient to apply.

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 legs:
• along the leg and hypotenuse: or
• along the leg and the 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 sharp corner: or
• from the proportionality of the 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 leg 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 leg to the adjacent one:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

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:

• through the catheters:
• through the leg and an acute angle: .

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

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

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's 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 the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will 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 much 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 exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

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

Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!

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“Understood” and “I know how to solve” are completely different skills. You need both.

Find problems and solve!

Side a can be identified as adjacent to corner B and opposite corner A, and the side b- as adjacent to corner A and opposite corner B.

Types of Right Triangles

• If the lengths of all three sides of a right triangle are integers, then the triangle is called Pythagorean triangle, and the lengths of its sides form the so-called Pythagorean triple.

Properties

Height

Height of a right triangle.

Trigonometric relations

Let be h and s (h>s) by the sides of two squares inscribed in a right triangle with a hypotenuse c. Then:

The perimeter of a right triangle is equal to the sum of the radii of the inscribed circle and the three circumscribed circles.

Notes

Links

• Weisstein, Eric W. Right Triangle (English) on the Wolfram MathWorld website.
• Wentworth G.A. A Text-Book of Geometry . - Ginn & Co., 1895.

Wikimedia Foundation. 2010 .

• cuboid
• Direct costs

See what "Right Triangle" is in other dictionaries:

right triangle- — Topics oil and gas industry EN right triangle … Technical Translator's Handbook

TRIANGLE- and (simple) triangle, triangle, husband. 1. A geometric figure bounded by three mutually intersecting straight lines forming three internal angles (mat.). Obtuse triangle. Acute triangle. Right triangle.… … Explanatory Dictionary of Ushakov

RECTANGULAR- RECTANGULAR, rectangular, rectangular (geom.). Having a right angle (or right angles). Right triangle. Rectangular figures. Explanatory Dictionary of Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary of Ushakov

Triangle- This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is a geometric figure formed by three line segments that connect three non-linear points. Three dots, ... ... Wikipedia

triangle- ▲ a polygon having, three, angle triangle is the simplest polygon; is given by 3 points that do not lie on the same straight line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language

TRIANGLE- A TRIANGLE, a, husband. 1. The geometric figure is a polygon with three corners, as well as any object, a device of this form. Rectangular t. Wooden t. (for drawing). Soldier's t. (soldier's letter without an envelope, folded in a corner; colloquial). 2… Explanatory dictionary of Ozhegov

Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, polygon with 3 sides. Sometimes under... Illustrated Encyclopedic Dictionary

triangle encyclopedic Dictionary

triangle- a; m. 1) a) A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles triangle/flax. Calculate the area of ​​the triangle. b) resp. what or with def. A figure or object of such a form. ... ... Dictionary of many expressions

Triangle- a; m. 1. A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles m. Calculate the area of ​​the triangle. // what or with def. A figure or object of such a form. T. roof. T.… … encyclopedic Dictionary

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 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. Triangle Properties

If the leg is opposite an angle of thirty degrees, then its length corresponds to half the length of the hypotenuse. It follows from this 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 mean proportional to the 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 \u003d c 2. The Pythagorean theorem is used to solve all geometric problems in which right triangles appear. 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 heights are combined with the legs. To find the third side, you need to find the sum of the projections of the legs on the hypotenuse and divide by two. If you draw a median from the vertex of a right angle, then it will turn out to be the radius of the circle that was described around the triangle. The center of this circle will be the midpoint 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 \u003d a * b / 2, which says 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. The sine is the ratio of the leg opposite the angle to the hypotenuse. Tangent is the ratio of the leg opposite the corner to the leg adjacent to this corner. The cotangent is the ratio of the leg adjacent to the corner to the leg opposite the corner. 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 the 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 legs, 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 height drawn from the right angle of an isosceles right triangle are the same.