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Formula of the Pythagorean theorem. Pythagorean theorem: history, proof, examples of practical application

The potential for creativity is usually attributed to the humanities, leaving the natural science to analysis, a practical approach and the dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity you won’t go far in the “queen of all sciences” - people have known this for a long time. Since the time of Pythagoras, for example.

School textbooks, unfortunately, usually do not explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and feel its fundamental principles. And at the same time, try to free your mind from cliches and elementary truths - only in such conditions are all great discoveries born.

Such discoveries include what we know today as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be exciting. And that this adventure is suitable not only for nerds with thick glasses, but for everyone who is strong in mind and strong in spirit.

From the history of the issue

Strictly speaking, although the theorem is called the “Pythagorean theorem,” Pythagoras himself did not discover it. The right triangle and its special properties were studied long before it. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.

Today you can no longer check who is right and who is wrong. What is known is that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid’s Elements may belong to Pythagoras, and Euclid only recorded it.

It is also known today that problems about a right triangle are found in Egyptian sources from the time of Pharaoh Amenemhat I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise “Sulva Sutra” and the ancient Chinese work “Zhou-bi suan jin”.

As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. This is confirmed by about 367 different pieces of evidence that exist today. In this, no other theorem can compete with it. Among the famous authors of proofs we can recall Leonardo da Vinci and the twentieth US President James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or are somehow connected with it.

Proofs of the Pythagorean theorem

School textbooks mostly give algebraic proofs. But the essence of the theorem is in geometry, so let’s first consider those proofs of the famous theorem that are based on this science.

Evidence 1

For the simplest proof of the Pythagorean theorem for a right triangle, you need to set ideal conditions: let the triangle be not only right-angled, but also isosceles. There is reason to believe that it was precisely this kind of triangle that ancient mathematicians initially considered.

Statement “a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs” can be illustrated with the following drawing:

Look at the isosceles right triangle ABC: On the hypotenuse AC, you can construct a square consisting of four triangles equal to the original ABC. And on sides AB and BC a square is built, each of which contains two similar triangles.

By the way, this drawing formed the basis of numerous jokes and cartoons dedicated to the Pythagorean theorem. The most famous is probably "Pythagorean pants are equal in all directions":

Evidence 2

This method combines algebra and geometry and can be considered a variant of the ancient Indian proof of the mathematician Bhaskari.

Construct a right triangle with sides a, b and c(Fig. 1). Then construct two squares with sides equal to the sum of the lengths of the two legs - (a+b). In each of the squares, make constructions as in Figures 2 and 3.

In the first square, build four triangles similar to those in Figure 1. The result is two squares: one with side a, the second with side b.

In the second square, four similar triangles constructed form a square with a side equal to the hypotenuse c.

The sum of the areas of the constructed squares in Fig. 2 is equal to the area of ​​the square we constructed with side c in Fig. 3. This can be easily checked by calculating the area of ​​the squares in Fig. 2 according to the formula. And the area of ​​the inscribed square in Figure 3. by subtracting the areas of four equal right triangles inscribed in the square from the area of ​​a large square with a side (a+b).

Writing all this down, we have: a 2 +b 2 =(a+b) 2 – 2ab. Open the brackets, carry out all the necessary algebraic calculations and get that a 2 +b 2 = a 2 +b 2. In this case, the area inscribed in Fig. 3. square can also be calculated using the traditional formula S=c 2. Those. a 2 +b 2 =c 2– you have proven the Pythagorean theorem.

Evidence 3

The ancient Indian proof itself was described in the 12th century in the treatise “The Crown of Knowledge” (“Siddhanta Shiromani”) and as the main argument the author uses an appeal addressed to the mathematical talents and observation skills of students and followers: “Look!”

But we will analyze this proof in more detail:

Inside the square, build four right triangles as indicated in the drawing. Let us denote the side of the large square, also known as the hypotenuse, With. Let's call the legs of the triangle A And b. According to the drawing, the side of the inner square is (a-b).

Use the formula for the area of ​​a square S=c 2 to calculate the area of ​​the outer square. And at the same time calculate the same value by adding the area of ​​the inner square and the areas of all four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options for calculating the area of ​​a square to make sure they give the same result. And this gives you the right to write down that c 2 =(a-b) 2 +4*1\2*a*b. As a result of the solution, you will receive the formula of the Pythagorean theorem c 2 =a 2 +b 2. The theorem is proven.

Proof 4

This curious ancient Chinese proof was called the “Bride’s Chair” - because of the chair-like figure that results from all the constructions:

It uses the drawing that we have already seen in Fig. 3 in the second proof. And the inner square with side c is constructed in the same way as in the ancient Indian proof given above.

If you mentally cut off two green right triangles from the drawing in Fig. 1, move them to opposite sides attach a square with side c and hypotenuses to the hypotenuses of lilac triangles, you will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will make sure that the “bride’s chair” is formed by two squares: small ones with a side b and big with a side a.

These constructions allowed the ancient Chinese mathematicians and us, following them, to come to the conclusion that c 2 =a 2 +b 2.

Evidence 5

This is another way to find a solution to the Pythagorean theorem using geometry. It's called the Garfield Method.

Construct a right triangle ABC. We need to prove that BC 2 = AC 2 + AB 2.

To do this, continue the leg AC and construct a segment CD, which is equal to the leg AB. Lower the perpendicular AD line segment ED. Segments ED And AC are equal. Connect the dots E And IN, and E And WITH and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tried: we find the area of ​​the resulting figure in two ways and equate the expressions to each other.

Find the area of ​​a polygon ABED can be done by adding up the areas of the three triangles that form it. And one of them, ERU, is not only rectangular, but also isosceles. Let's also not forget that AB=CD, AC=ED And BC=SE– this will allow us to simplify the recording and not overload it. So, S ABED =2*1/2(AB*AC)+1/2ВС 2.

At the same time, it is obvious that ABED- This is a trapezoid. Therefore, we calculate its area using the formula: S ABED =(DE+AB)*1/2AD. For our calculations, it is more convenient and clearer to represent the segment AD as the sum of segments AC And CD.

Let's write down both ways to calculate the area of ​​a figure, putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify the right side of the notation: AB*AC+1/2BC 2 =1/2(AB+AC) 2. Now let’s open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having completed all the transformations, we get exactly what we need: BC 2 = AC 2 + AB 2. We have proven the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proven using vectors, complex numbers, differential equations, stereometry, etc. And even physicists: if, for example, liquid is poured into square and triangular volumes similar to those shown in the drawings. By pouring liquid, you can prove the equality of areas and the theorem itself as a result.

A few words about Pythagorean triplets

This issue is little or not studied at all in the school curriculum. Meanwhile, he is very interesting and has great importance in geometry. Pythagorean triples are used to solve many mathematical problems. Understanding them may be useful to you in further education.

So what are Pythagorean triplets? That's what they call it integers, collected in threes, the sum of the squares of two of which is equal to the third number in the square.

Pythagorean triples can be:

• primitive (all three numbers are relatively prime);
• not primitive (if each number of a triple is multiplied by the same number, you get a new triple, which is not primitive).

Even before our era, the ancient Egyptians were fascinated by the mania for numbers of Pythagorean triplets: in problems they considered a right triangle with sides of 3, 4 and 5 units. By the way, any triangle whose sides are equal to the numbers from the Pythagorean triple is rectangular by default.

Examples of Pythagorean triplets: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20 ), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (10, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (27, 36, 45), (14 , 48, 50), (30, 40, 50), etc.

Practical application of the theorem

The Pythagorean theorem is used not only in mathematics, but also in architecture and construction, astronomy and even literature.

First about construction: the Pythagorean theorem finds in it wide application in tasks different levels difficulties. For example, look at a Romanesque window:

Let us denote the width of the window as b, then the radius of the major semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed through b: r=b/4. In this problem we are interested in the radius of the inner circle of the window (let's call it p).

The Pythagorean theorem is just useful to calculate R. To do this, we use a right triangle, which is indicated by a dotted line in the figure. The hypotenuse of a triangle consists of two radii: b/4+p. One leg represents the radius b/4, another b/2-p. Using the Pythagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Next, we open the brackets and get b 2 /16+ bp/2+p 2 =b 2 /16+b 2 /4-bp+p 2. Let's transform this expression into bp/2=b 2 /4-bp. And then we divide all terms by b, we present similar ones to get 3/2*p=b/4. And in the end we find that p=b/6- which is what we needed.

Using the theorem, you can calculate the length of the rafters for gable roof. Determine how high a mobile communications tower is needed for the signal to reach a certain populated area. And even install steadily christmas tree on the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

In literature, the Pythagorean theorem has inspired writers since antiquity and continues to do so in our time. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired to write a sonnet:

The light of truth will not dissipate soon,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
It will not cause doubt or controversy.

The wisest when it touches your gaze
Light of truth, thank the gods;
And a hundred bulls, slaughtered, lie -
A return gift from the lucky Pythagoras.

Since then the bulls have been roaring desperately:
Forever alarmed the bull tribe
Event mentioned here.

It seems to them that the time is about to come,
And they will be sacrificed again
Some great theorem.

(translation by Viktor Toporov)

And in the twentieth century, the Soviet writer Evgeniy Veltistov, in his book “The Adventures of Electronics,” devoted an entire chapter to proofs of the Pythagorean theorem. And another half chapter to the story about the two-dimensional world that could exist if the Pythagorean theorem became a fundamental law and even a religion for a single world. Living there would be much easier, but also much more boring: for example, no one there understands the meaning of the words “round” and “fluffy”.

And in the book “The Adventures of Electronics,” the author, through the mouth of mathematics teacher Taratar, says: “The main thing in mathematics is the movement of thought, new ideas.” It is precisely this creative flight of thought that gives rise to the Pythagorean theorem - it is not for nothing that it has so many varied proofs. It helps you go beyond the boundaries of the familiar and look at familiar things in a new way.

Conclusion

This article was created so that you can look beyond the school curriculum in mathematics and learn not only those proofs of the Pythagorean theorem that are given in the textbooks “Geometry 7-9” (L.S. Atanasyan, V.N. Rudenko) and “Geometry 7” -11” (A.V. Pogorelov), but also other interesting ways to prove the famous theorem. And also see examples of how the Pythagorean theorem can be applied in everyday life.

Firstly, this information will allow you to qualify for higher scores in mathematics lessons - information on the subject from additional sources are always highly appreciated.

Secondly, we wanted to help you feel how interesting mathematics is. Make sure specific examples that there is always a place for creativity in it. We hope that the Pythagorean theorem and this article will inspire you to independently explore and make exciting discoveries in mathematics and other sciences.

Tell us in the comments if you found the evidence presented in the article interesting. Did you find this information useful in your studies? Write to us what you think about the Pythagorean theorem and this article - we will be happy to discuss all this with you.

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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(the one and only, the unique and the 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 proven 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:

Square of the hypotenuse equal to the sum squares of 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 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 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 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 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 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 square MKOR 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 12 . 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. Carry out in given square diagonals. 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.)

When you first started learning about square roots and how to solve irrational equations (equalities involving an unknown under the root sign), you probably got your first taste of them. practical use. Ability to extract Square root from numbers is also necessary to solve problems using the Pythagorean theorem. This theorem relates the lengths of the sides of any right triangle.

Let the lengths of the legs of a right triangle (those two sides that meet at right angles) be designated by the letters and, and the length of the hypotenuse (the longest side of the triangle located opposite the right angle) will be designated by the letter. Then the corresponding lengths are related by the following relation:

This equation allows you to find the length of a side of a right triangle when the length of its other two sides is known. In addition, it allows you to determine whether the triangle in question is a right triangle, provided that the lengths of all three sides are known in advance.

Solving problems using the Pythagorean theorem

To consolidate the material, we will solve the following problems using the Pythagorean theorem.

So, given:

1. The length of one of the legs is 48, the hypotenuse is 80.
2. The length of the leg is 84, the hypotenuse is 91.

Let's get to the solution:

a) Substituting the data into the above equation gives the following results:

48 2 + b 2 = 80 2

2304 + b 2 = 6400

b 2 = 4096

b= 64 or b = -64

Since the length of a side of a triangle cannot be expressed negative number, the second option is automatically discarded.

Answer to the first picture: b = 64.

b) The length of the leg of the second triangle is found in the same way:

84 2 + b 2 = 91 2

7056 + b 2 = 8281

b 2 = 1225

b= 35 or b = -35

As in the previous case, negative decision discarded.

Answer to the second picture: b = 35

We are given:

1. The lengths of the smaller sides of the triangle are 45 and 55, respectively, and the larger sides are 75.
2. The lengths of the smaller sides of the triangle are 28 and 45, respectively, and the larger sides are 53.

Let's solve the problem:

a) It is necessary to check whether the sum of the squares of the lengths of the shorter sides of a given triangle is equal to the square of the length of the larger:

45 2 + 55 2 = 2025 + 3025 = 5050

Therefore, the first triangle is not a right triangle.

b) The same operation is performed:

28 2 + 45 2 = 784 + 2025 = 2809

Therefore, the second triangle is a right triangle.

First, let's find the length of the largest segment formed by points with coordinates (-2, -3) and (5, -2). To do this, we use the well-known formula for finding the distance between points in a rectangular coordinate system:

Similarly, we find the length of the segment enclosed between points with coordinates (-2, -3) and (2, 1):

Finally, we determine the length of the segment between points with coordinates (2, 1) and (5, -2):

Since the equality holds:

then the corresponding triangle is right-angled.

Thus, we can formulate the answer to the problem: since the sum of the squares of the sides with the shortest length is equal to the square of the side with the longest length, the points are the vertices of a right triangle.

The base (located strictly horizontally), the jamb (located strictly vertically) and the cable (stretched diagonally) form a right triangle, respectively, to find the length of the cable the Pythagorean theorem can be used:

Thus, the length of the cable will be approximately 3.6 meters.

Given: the distance from point R to point P (the leg of the triangle) is 24, from point R to point Q (hypotenuse) is 26.

So, let’s help Vita solve the problem. Since the sides of the triangle shown in the figure are supposed to form a right triangle, you can use the Pythagorean theorem to find the length of the third side:

So, the width of the pond is 10 meters.

Sergey Valerievich

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, lying opposite the right angle). 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 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.

Take the square root of both sides of the equation after you have the unknown (squared) on one side of the equation and the intercept (a number) 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. The bottom of the stairs is 5 meters from the base of the wall. Top part
  • “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.