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).

When did you first start learning about square roots and how to solve them? irrational equations(equalities containing an unknown under the root sign), you probably got the first idea about them practical use. The ability to take the square root of 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 right-angled, provided that the lengths of all three sides 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 is equal given triangle the square of the greater length:

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). For this we use well-known formula to find the distance between points in rectangular system coordinates:

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 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

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 ones Pythagorean pants and let's 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 be glad that we have simple wording Pythagorean theorem. Let's repeat it again to remember it better:

It should be easy now:

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

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

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

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

1.
Actually it sounds like this:

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

Sine of an acute angle equal to the ratio 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 the usual signs of equality of triangles? 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 all angles are equal!

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:

Pythagorean theorem

Proof

On at the moment V scientific literature 367 proofs of this theorem have been recorded. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. Such diversity can only be explained by the fundamental significance of the theorem for geometry.
Of course, conceptually all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs (for example, using differential equations).

Through similar triangles

The following proof of the algebraic formulation is the simplest of the proofs, constructed directly from the axioms. In particular, it does not use the concept of area of ​​a figure.
Let ABC be a right triangle with right angle C. Draw the altitude from C and denote its base by H. Triangle ACH is similar to triangle ABC at two angles.
Similarly, triangle CBH is similar to ABC. By introducing the notation

we get

What is equivalent

Adding it up, we get

or

Proofs using the area method

The proofs below, despite their apparent simplicity, are not so simple at all. They all use properties of area, the proof of which is more complex than the proof of the Pythagorean theorem itself.

Proof via equicomplementation

Proofs through equivalence

An example of one such proof is shown in the drawing on the right, where a square built on the hypotenuse is rearranged into two squares built on the legs.

Euclid's proof

Proof of Leonardo da Vinci

The main elements of the proof are symmetry and motion.

Let's consider the drawing, as can be seen from the symmetry, the segment CI cuts the square ABHJ into two identical parts (since triangles ABC and JHI are equal in construction). Using a 90 degree counterclockwise rotation, we see the equality of the shaded figures CAJI and GDAB. Now it is clear that the area of ​​the figure we have shaded is equal to the sum of half the areas of the squares built on the legs and the area of ​​the original triangle. On the other hand, it is equal to half the area of ​​the square built on the hypotenuse, plus the area of ​​the original triangle. The last step in the proof is left to the reader.

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relation

between the sides of a right triangle.

It is believed that it was proven by the Greek mathematician Pythagoras, after whom it was named.

Geometric formulation of the Pythagorean theorem.

The theorem was originally formulated as follows:

In a right triangle, the area of ​​the square built on the hypotenuse is equal to the sum of the areas of the squares,

built on legs.

Algebraic formulation of the Pythagorean theorem.

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

That is, denoting the length of the hypotenuse of the triangle by c, and the lengths of the legs through a And b:

Both formulations Pythagorean theorem are equivalent, but the second formulation is more elementary, it does not

requires the concept of area. That is, the second statement can be verified without knowing anything about the area and

by measuring only the lengths of the sides of a right triangle.

Converse Pythagorean theorem.

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then

right triangle.

Or, in other words:

For every triple of positive numbers a, b And c, such that

there is a right triangle with legs a And b and hypotenuse c.

Pythagorean theorem for an isosceles triangle.

Pythagorean theorem for an equilateral triangle.

Proofs of the Pythagorean theorem.

Currently, 367 proofs of this theorem have been recorded in the scientific literature. Probably the theorem

Pythagoras is the only theorem with such an impressive number of proofs. Such diversity

can only be explained by the fundamental significance of the theorem for geometry.

Of course, conceptually all of them can be divided into a small number of classes. The most famous of them:

proof area method, axiomatic And exotic evidence(For example,

by using differential equations).

1. Proof of the Pythagorean theorem using similar triangles.

The following proof of the algebraic formulation is the simplest of the proofs constructed

directly from the axioms. In particular, it does not use the concept of area of ​​a figure.

Let ABC there is a right triangle with a right angle C. Let's draw the height from C and denote

its foundation through H.

Triangle ACH similar to a triangle AB C at two corners. Likewise, triangle CBH similar ABC.

By introducing the notation:

we get:

,

which corresponds to -

Folded a 2 and b 2, we get:

or , which is what needed to be proven.

2. Proof of the Pythagorean theorem using the area method.

The proofs below, despite their apparent simplicity, are not so simple at all. All of them

use properties of area, the proofs of which are more complex than the proof of the Pythagorean theorem itself.

• Proof through equicomplementarity.

Let's arrange four equal rectangular

triangle as shown in the figure

right.

Quadrangle with sides c- square,

since the sum of two acute angles is 90°, and

unfolded angle - 180°.

The area of ​​the entire figure is, on the one hand,

area of ​​a square with side ( a+b), and on the other hand, the amount four squares triangles and

Q.E.D.

3. Proof of the Pythagorean theorem by the infinitesimal method.

​

Looking at the drawing shown in the figure and

watching the side changea, we can

write the following relation for infinitely

small side incrementsWith And a(using similarity

triangles):

Using the variable separation method, we find:

More general expression to change the hypotenuse in case of increments of both legs:

Integrating given equation and using initial conditions, we get:

Thus we arrive at the desired answer:

As is easy to see, the quadratic dependence in the final formula appears due to the linear

proportionality between the sides of the triangle and the increments, while the sum is related to the independent

contributions from the increment of different legs.

A simpler proof can be obtained if we assume that one of the legs does not experience an increase

(V in this case leg b). Then for the integration constant we obtain: