All about right triangles. Right triangle

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

This concept means consisting of three angles and

sides, with one of the angles measuring 90 degrees. The sides that make up a right angle are called legs, and the third side, which is opposite to 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 membership in two, which means that the properties of both groups are observed. Let us remember 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.

Availability of one of following properties allows us to state that one right triangle is equal to another:

the sides of two triangles are equal;
the figures have the same hypotenuse and one of the legs;
the hypotenuse and any of the acute angles are equal;
the condition of equality of the leg and the acute angle is met.

The area of a right triangle is 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:

the leg is nothing more than the mean proportional to the hypotenuse and its projection onto it;
if you describe a circle around a right triangle, its center will be in the middle of the hypotenuse;
height drawn from right angle, represents the average proportional with the projections of the legs of the triangle onto its hypotenuse.

The interesting thing is that no matter what the right triangle is, these properties are always respected.

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 relationship when he was studying the properties of squares built on

To prove the theorem, we construct a triangle ABC, the legs of which we denote as a and b, and the hypotenuse as c. Next we will build two squares. For one, the side will be the hypotenuse, for 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 four triangles ABC and the second square, or as the square of the side; naturally, these ratios will be equal. That is:

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 = a 2 + b 2

Thus, geometric figure a right triangle not only corresponds 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 will be useful not only in science, but also in everyday life, since such a figure as a right triangle is found everywhere.

Instructions

The angles opposite to legs a and b will be denoted by A and B, respectively. The hypotenuse, by definition, is the side of a right triangle that is opposite to the right angle (while the hypotenuse forms acute angles with the other sides of the triangle). We denote the length of the hypotenuse by c.

You will need:
Calculator.

Use the following expression for the leg: a=sqrt(c^2-b^2), if you know the values of the hypotenuse and the other leg. This expression is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle is the sum of the squares of the legs. The sqrt operator extracts square roots. The sign "^2" means raising to the second power.

Use the formula a=c*sinA if you know the hypotenuse (c) and the angle opposite to the desired one (we denoted this angle as A).
Use the expression a=c*cosB to find a leg if you know the hypotenuse (c) and the angle adjacent to the desired leg (we denoted this angle as B).
Calculate the leg from a=b*tgA in the case where leg b and the angle opposite to the desired leg are given (we agreed to denote this angle as A).

Please note:
If the leg in your problem is not found in any of the described ways, most likely it can be reduced to one of them.

Useful tips:
All these expressions are obtained from well-known definitions trigonometric functions, therefore, even if you forgot one of them, you can always quickly retrieve it through simple operations. It is also useful to know the values of trigonometric functions for the most common angles of 30, 45, 60, 90, 180 degrees.

Video on the topic

Sources:

“A manual on mathematics for those entering universities,” ed. G.N. Yakovleva, 1982
leg of a right triangle

A square triangle is more accurately called a right triangle. The relationships between the sides and angles of this geometric figure are discussed in detail in the mathematical discipline of trigonometry.

You will need

- a sheet of paper;
- pen;
- Bradis tables;
- calculator.

Instructions

Find triangle using the Pythagorean theorem. According to this theorem, the square of the hypotenuse is equal to the sum of the squares of the legs: c2 = a2+b2, where c is the hypotenuse triangle, a and b are its legs. To apply this, you need to know the length of any two sides of the rectangular triangle.

If the conditions specify the dimensions of the legs, find the length of the hypotenuse. To do this, use square root from the sum of the legs, each of which must first be squared.

Calculate the length of one of the legs if the dimensions of the hypotenuse and the other leg are known. Using a calculator, take the square root of the difference between the hypotenuse and famous leg, also squared.

If the problem specifies the hypotenuse and one of the acute angles adjacent to it, use Bradis tables. They provide the values of trigonometric functions for large number corners Use a calculator with sine and cosine functions, as well as trigonometry theorems that describe the relationships between sides and rectangular triangle.

Find the legs using basic trigonometric functions: a = c*sin α, b = c*cos α, where a is the leg opposite to angle α, b is the leg adjacent to angle α. Calculate the size of the sides in the same way triangle, if the hypotenuse and another acute angle are given: b = c*sin β, a = c*cos β, where b is the leg opposite to angle β, and is the leg adjacent to angle β.

In the case of a and an adjacent acute angle β, do not forget that in a right triangle the sum of the acute angles is always equal to 90°: α + β = 90°. Find the value of the angle opposite to leg a: α = 90° – β. Or use trigonometric reduction formulas: sin α = sin (90° – β) = cos β; tan α = tan (90° – β) = ctg β = 1/tg β.

Video on the topic

Sources:

How to find the sides of a right triangle by leg and acute angle in 2019

Tip 3: How to find an acute angle in a right triangle

Directly carbonic the triangle is probably one of the most famous, from a historical point of view, geometric figures. Pythagorean “pants” can only compete with “Eureka!” Archimedes.

You will need

- drawing of a triangle;
- ruler;
- protractor

Instructions

The sum of the angles of a triangle is 180 degrees. In a rectangular triangle one angle (straight) will always be 90 degrees, and the rest are acute, i.e. less than 90 degrees each. To determine what angle is in a rectangular triangle is straight, use a ruler to measure the sides of the triangle and determine the largest. It is the hypotenuse (AB) and is located opposite the right angle (C). The remaining two sides form a right angle and legs (AC, BC).

Once you have determined which angle is acute, you can either use a protractor to calculate the angle using mathematical formulas.

To determine the angle using a protractor, align its top (let’s denote it with the letter A) with a special mark on the ruler in the center of the protractor; leg AC should coincide with its upper edge. Mark on the semicircular part of the protractor the point through which the hypotenuse AB. The value at this point corresponds to the angle in degrees. If there are 2 values indicated on the protractor, then for an acute angle you need to choose the smaller one, for an obtuse angle - the larger one.

Find the resulting value in the Bradis reference books and determine which angle the resulting value corresponds to numeric value. Our grandmothers used this method.

In our case, it is enough to take with the calculation function trigonometric formulas. For example, the built-in Windows calculator. Launch the "Calculator" application, in the "View" menu item, select "Engineering". Calculate the sine of the desired angle, for example, sin (A) = BC/AB = 2/4 = 0.5

Switch the calculator to inverse function mode by clicking on the INV button on the calculator display, then click on the arcsine function button (indicated on the display as sin minus the first power). The following message will appear in the calculation window: asind (0.5) = 30. I.e. the value of the desired angle is 30 degrees.

In life we will often have to deal with math problems: at school, at university, and then helping your child with completing homework. People in certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article we will look at one of them: finding the side of a right triangle.

What is a right triangle

First, let's remember what a right triangle is. A right triangle is a geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides forming a right angle are called legs, and the side that lies opposite the right angle is called the hypotenuse.

Finding the leg of a right triangle

There are several ways to find out the length of the leg. I would like to consider them in more detail.

Pythagorean theorem to find the side of a right triangle

If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².

Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next we solve: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).

Trigonometric ratios to find the leg of a right triangle

You can also find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. The table below will help us solve problems. Let's consider these options.

Find the leg of a right triangle using sine

The sine of an angle (sin) is the ratio of the opposite side to the hypotenuse. Formula: sin=a/c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.

Example. The hypotenuse is 10 cm, angle A is 30 degrees. Using the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).

Find the leg of a right triangle using cosine

The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos=b/c, where b is the leg adjacent to a given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.

Example. Angle A is equal to 60 degrees, the hypotenuse is equal to 10 cm. Using the table, we calculate the cosine of angle A, it is equal to 1/2. Next we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).

Find the leg of a right triangle using tangent

Tangent of an angle (tg) is the ratio of the opposite side to the adjacent side. Formula: tg=a/b, where a is the side opposite to the angle, and b is the adjacent side. Let's transform the formula and get: a=tg*b.

Example. Angle A is equal to 45 degrees, the hypotenuse is equal to 10 cm. Using the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).

Find the leg of a right triangle using cotangent

Angle cotangent (ctg) is the ratio of the adjacent side to the opposite side. Formula: ctg=b/a, where b is the leg adjacent to the angle, and is the opposite leg. In other words, cotangent is an “inverted tangent.” We get: b=ctg*a.

Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. We calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).

So now you know how to find a leg in a right triangle. As you can see, it’s not that difficult, the main thing is to remember the formulas.

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. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. And the best thing about it is that it is simple.

So, Pythagorean theorem:

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

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

Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

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

Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.

In this picture, the sum of the areas of the small squares is equal to the area of the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem?

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:

It should be easy now:

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 lengths and!

Now let’s connect the marked dots

Here we, however, noted something else, but you yourself look at the drawing and think why this is so.

What is the area of the larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” is equal.

Let's put it all together now.

Let's convert:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite side to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.

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

And once again all this in the form of a tablet:

It'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

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 do you know about the diagonals of a rectangle?

And what follows from this?

So it turned out that

- 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: