Make sure that the triangle you are given is a right triangle, as the Pythagorean Theorem only applies to right triangles. In right triangles, one of the three angles is always 90 degrees.

• A right angle in a right triangle is indicated by a square symbol, rather than by the curve symbol that represents oblique angles.

Label the sides of the triangle. Label the legs as “a” and “b” (legs are sides intersecting at right angles), and the hypotenuse as “c” (the hypotenuse is the most big side right triangle, opposite right angle).

Determine which side of the triangle you want to find. The Pythagorean theorem allows you to find any side of a right triangle (if the other two sides are known). Determine which side (a, b, c) you need to find.

• For example, given a hypotenuse equal to 5, and given a leg equal to 3. In this case, it is necessary to find the second leg. We'll come back to this example later.
• If the other two sides are unknown, you need to find the length of one of the unknown sides to be able to apply the Pythagorean theorem. To do this, use the basic trigonometric functions(if you are given the value of one of the oblique angles).

Substitute the values ​​given to you (or the values ​​you found) into the formula a 2 + b 2 = c 2. Remember that a and b are legs, and c is the hypotenuse.

• In our example, write: 3² + b² = 5².

Square each known side. Or leave the powers - you can square the numbers later.

• In our example, write: 9 + b² = 25.

Isolate the unknown side on one side of the equation. To do this, move known values to the other side of the equation. If you find the hypotenuse, then in the Pythagorean theorem it is already isolated on one side of the equation (so you don't need to do anything).

• In our example, move 9 to right side equations to isolate the unknown b². You will get b² = 16.

Remove square root from both sides of the equation after the unknown (squared) is present on one side of the equation and the free term (number) is present on the other side.

• In our example, b² = 16. Take the square root of both sides of the equation and get b = 4. Thus, the second leg is 4.

Use the Pythagorean theorem in everyday life, since it can be used in large number practical situations. To do this, learn to recognize right triangles in everyday life - in any situation in which two objects (or lines) intersect at right angles, and a third object (or line) connects (diagonally) the tops of the first two objects (or lines), you can use the Pythagorean theorem to find the unknown side (if the other two sides are known).

• Example: given a staircase leaning against a building. Bottom part The stairs are located 5 meters from the base of the wall. Upper part The stairs are located 20 meters from the ground (up the wall). What is the length of the stairs?
  • “5 meters from the base of the wall” means that a = 5; “located 20 meters from the ground” means that b = 20 (that is, you are given two legs of a right triangle, since the wall of the building and the surface of the Earth intersect at right angles). The length of the staircase is the length of the hypotenuse, which is unknown.
    • a² + b² = c²
    • (5)² + (20)² = c²
    • 25 + 400 = c²
    • 425 = c²
    • c = √425
    • c = 20.6. Thus, the approximate length of the stairs is 20.6 meters.

Geometry is not a simple science. It may be useful for both school curriculum, and in real life. Knowledge of many formulas and theorems will simplify geometric calculations. One of the most simple figures in geometry it is a triangle. One of the varieties of triangles, equilateral, has its own characteristics.

Features of an equilateral triangle

By definition, a triangle is a polyhedron that has three angles and three sides. This is a flat two-dimensional figure, its properties are studied in high school. Based on the type of angle, there are acute, obtuse and right triangles. Right triangle- such geometric figure, where one of the angles is 90º. Such a triangle has two legs (they create a right angle) and one hypotenuse (it is opposite the right angle). Depending on what quantities are known, there are three simple ways Calculate the hypotenuse of a right triangle.

The first way is to find the hypotenuse of a right triangle. Pythagorean theorem

The Pythagorean theorem is the oldest way to calculate any of the sides of a right triangle. It sounds like this: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.” Thus, to calculate the hypotenuse, one must derive the square root of the sum of two legs squared. For clarity, formulas and a diagram are given.

Second way. Calculation of the hypotenuse using 2 known quantities: leg and adjacent angle

One of the properties of a right triangle states that the ratio of the length of the leg to the length of the hypotenuse is equivalent to the cosine of the angle between this leg and the hypotenuse. Let's call the angle known to us α. Now, thanks to the well-known definition, you can easily formulate a formula for calculating the hypotenuse: Hypotenuse = leg/cos(α)

Third way. Calculation of the hypotenuse using 2 known quantities: leg and opposite angle

If the opposite angle is known, it is possible to again use the properties of a right triangle. The ratio of the length of the leg and the hypotenuse is equivalent to the sine of the opposite angle. Let us again call the known angle α. Now for the calculations we will use a slightly different formula:
Hypotenuse = leg/sin (α)

Examples to help you understand formulas

For a deeper understanding of each of the formulas, you should consider illustrative examples. So, suppose you are given a right triangle, where there is the following data:

• Leg – 8 cm.
• The adjacent angle cosα1 is 0.8.
• The opposite angle sinα2 is 0.8.

According to the Pythagorean theorem: Hypotenuse = square root of (36+64) = 10 cm.
According to the size of the leg and adjacent angle: 8/0.8 = 10 cm.
According to the size of the leg and the opposite angle: 8/0.8 = 10 cm.

Once you understand the formula, you can easily calculate the hypotenuse with any data.

Video: Pythagorean Theorem

Intermediate level

Right triangle. The Complete Illustrated Guide (2019)

RECTANGULAR TRIANGLE. ENTRY 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. 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:

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

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 there is! 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:

Resume

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 segments of 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 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:

Sinus acute angle 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's very convenient!

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.

It is necessary that 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 usual signs triangle congruence? 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 is known 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:

MEASUREMENT OF AREA OF GEOMETRIC FIGURES.

§ 58. PYTHAGOREAN THEOREM 1.

__________
1 Pythagoras is a Greek scientist who lived about 2500 years ago (564-473 BC).
_________

Let us be given a right triangle whose sides A, b And With(drawing 267).

Let's build squares on its sides. The areas of these squares are respectively equal A 2 , b 2 and With 2. Let's prove that With 2 = a 2 + b 2 .

Let's construct two squares MKOR and M"K"O"R" (drawings 268, 269), taking as the side of each of them a segment equal to the sum of the legs of the right triangle ABC.

Having completed the constructions shown in drawings 268 and 269 in these squares, we will see that the MCOR square is divided into two squares with areas A 2 and b 2 and four equal right triangles, each of which is equal to right triangle ABC. The square M"K"O"R" was divided into a quadrangle (it is shaded in drawing 269) and four right triangles, each of which is also equal to triangle ABC. A shaded quadrilateral is a square, since its sides are equal (each is equal to the hypotenuse of triangle ABC, i.e. With), and the angles are right / 1 + / 2 = 90°, from where / 3 = 90°).

Thus, the sum of the areas of the squares built on the legs (in drawing 268 these squares are shaded) is equal to the area of ​​the MCOR square without the sum of the areas of four equal triangles, and the area of ​​the square built on the hypotenuse (in drawing 269 this square is also shaded) is equal to the area of ​​the square M"K"O"R", equal to the square MCOR, without the sum of the areas of four similar triangles. Therefore, the area of ​​a square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on the legs.

We get the formula With 2 = a 2 + b 2 where With- hypotenuse, A And b- legs of a right triangle.

The Pythagorean theorem is usually formulated briefly as follows:

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

From the formula With 2 = a 2 + b 2 you can get the following formulas:

A 2 = With 2 - b 2 ;
b 2 = With 2 - A 2 .

These formulas can be used to find the unknown side of a right triangle from its two given sides.
For example:

a) if the legs are given A= 4 cm, b=3 cm, then you can find the hypotenuse ( With):
With 2 = a 2 + b 2, i.e. With 2 = 4 2 + 3 2 ; with 2 = 25, whence With= √25 =5 (cm);

b) if the hypotenuse is given With= 17 cm and leg A= 8 cm, then you can find another leg ( b):

b 2 = With 2 - A 2, i.e. b 2 = 17 2 - 8 2 ; b 2 = 225, from where b= √225 = 15 (cm).

Consequence: If two right triangles ABC and A have 1 B 1 C 1 hypotenuse With And With 1 are equal, and leg b triangle ABC is longer than the leg b 1 triangle A 1 B 1 C 1,
then the leg A triangle ABC is smaller than the leg A 1 triangle A 1 B 1 C 1. (Make a drawing illustrating this consequence.)

In fact, based on the Pythagorean theorem we obtain:

A 2 = With 2 - b 2 ,
A 1 2 = With 1 2 - b 1 2

In the written formulas, the minuends are equal, and the subtrahend in the first formula is greater than the subtrahend in the second formula, therefore, the first difference is less than the second,
i.e. A 2 < A 1 2 . Where A< A 1 .

Exercises.

1. Using drawing 270, prove the Pythagorean theorem for an isosceles right triangle.

2. One leg of a right triangle is 12 cm, the other is 5 cm. Calculate the length of the hypotenuse of this triangle.

3. The hypotenuse of a right triangle is 10 cm, one of the legs is 8 cm. Calculate the length of the other leg of this triangle.

4. The hypotenuse of a right triangle is 37 cm, one of its legs is 35 cm. Calculate the length of the other leg of this triangle.

5. Construct a square with an area twice the size of the given one.

6. Construct a square with an area half the size of the given one. Note. Draw diagonals in this square. The squares constructed on the halves of these diagonals will be the ones we are looking for.

7. The legs of a right triangle are respectively 12 cm and 15 cm. Calculate the length of the hypotenuse of this triangle with an accuracy of 0.1 cm.

8. The hypotenuse of a right triangle is 20 cm, one of its legs is 15 cm. Calculate the length of the other leg to the nearest 0.1 cm.

9. How long must the ladder be so that it can be attached to a window located at a height of 6 m, if the lower end of the ladder must be 2.5 m from the building? (Chart 271.)

Every schoolchild knows that the square of the hypotenuse is always equal to the sum of the legs, each of which is squared. This statement is called the Pythagorean theorem. It is one of the most famous theorems of trigonometry and mathematics in general. Let's take a closer look at it.

The concept of a right triangle

Before moving on to consider the Pythagorean theorem, in which the square of the hypotenuse is equal to the sum of the legs that are squared, we should consider the concept and properties of a right triangle for which the theorem is valid.

Triangle - flat figure having three angles and three sides. A right triangle, as its name suggests, has one right angle, that is, this angle is equal to 90 o.

From general properties for all triangles, it is known that the sum of all three angles of this figure is 180 o, which means that for a right triangle, the sum of two angles that are not right angles is 180 o - 90 o = 90 o. This last fact means that any angle in a right triangle that is not right will always be less than 90 o.

The side that lies opposite the right angle is called the hypotenuse. The other two sides are the legs of the triangle, they can be equal to each other, or they can be different. From trigonometry we know that the greater the angle against which a side of a triangle lies, the greater the length of that side. This means that in a right triangle the hypotenuse (lies opposite the 90 o angle) will always be greater than any of the legs (lie opposite the angles< 90 o).

Mathematical notation of Pythagorean theorem

This theorem states that the square of the hypotenuse is equal to the sum of the legs, each of which is previously squared. To write this formulation mathematically, consider a right triangle in which sides a, b and c are the two legs and the hypotenuse, respectively. In this case, the theorem, which is formulated as the square of the hypotenuse is equal to the sum of the squares of the legs, can be represented by the following formula: c 2 = a 2 + b 2. From here other formulas important for practice can be obtained: a = √(c 2 - b 2), b = √(c 2 - a 2) and c = √(a 2 + b 2).

Note that in the case of a right-angled equilateral triangle, that is, a = b, the formulation: the square of the hypotenuse is equal to the sum of the legs, each of which is squared, will be mathematically written as follows: c 2 = a 2 + b 2 = 2a 2, which implies the equality: c = a√2.

Historical background

The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the legs, each of which is squared, was known long before the famous Greek philosopher paid attention to it. Many papyri Ancient Egypt, as well as clay tablets of the Babylonians confirm that these peoples used the noted property of the sides of a right triangle. For example, one of the first Egyptian pyramids, the Pyramid of Khafre, the construction of which dates back to the 26th century BC (2000 years before the life of Pythagoras), was built based on knowledge of the aspect ratio in a right triangle 3x4x5.

Why then does the theorem now bear the name of the Greek? The answer is simple: Pythagoras is the first to mathematically prove this theorem. The surviving Babylonian and Egyptian written sources only speak of its use, but do not provide any mathematical proof.

It is believed that Pythagoras proved the theorem in question by using the properties of similar triangles, which he obtained by drawing the height in a right triangle from an angle of 90 o to the hypotenuse.

An example of using the Pythagorean theorem

Let's consider simple task: it is necessary to determine the length of the inclined staircase L, if it is known that it has a height H = 3 meters, and the distance from the wall against which the staircase rests to its foot is P = 2.5 meters.

IN in this case H and P are the legs, and L is the hypotenuse. Since the length of the hypotenuse is equal to the sum of the squares of the legs, we get: L 2 = H 2 + P 2, from where L = √(H 2 + P 2) = √(3 2 + 2.5 2) = 3.905 meters or 3 m and 90, 5 cm.