The height from the right angle is equal. Right triangle

Middle level

Right triangle. Complete illustrated guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

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

and in such

and in such

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

Attention to the drawing!

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

Well, we discussed the names, now the most important thing: the Pythagorean Theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought many benefits to those who know it. And the best thing about her is that she is simple.

So, Pythagorean theorem:

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

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

Does it really look like shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

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

Doesn't it sound a little different, doesn't it? And so, when Pythagoras drew the statement of his theorem, just such a picture turned out.

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

Why are we now formulating the Pythagorean theorem

Did Pythagoras suffer and talk about squares?

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

Now it should be easy:

Well, the most important theorem about a right triangle was discussed. If you are interested in how it is proved, read the next levels of theory, and now let's move on ... into the dark forest ... of trigonometry! To the terrible words sine, cosine, tangent and cotangent.

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

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

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

1.
It actually sounds like this:

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

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

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

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

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

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

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

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

Now let's connect the marked points

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

What is the area of ​​the larger square?

Correctly, .

What about the smaller area?

Certainly, .

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

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

Let's put it all together now.

Let's transform:

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

Right triangle and trigonometry

For a right triangle, the following relations hold:

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

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

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

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

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

It is very comfortable!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

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

THEN THE TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both - opposite.

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

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

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

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

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it happened that

1. - median:

Remember this fact! Helps a lot!

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

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

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

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

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

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

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

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

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

Let's write them down again.

Pythagorean theorem:

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

Signs of equality of right triangles:

• on two legs:
• along the leg and hypotenuse: or
• along the leg and the adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one sharp corner: or
• from the proportionality of the two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent one:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

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

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

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

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

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

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

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

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

Find problems and solve!

Right triangle is a triangle in which one of the angles is right, that is, equal to 90 degrees.

• The side opposite the right angle is called the hypotenuse. c or AB)
• The side adjacent to the right angle is called the leg. Each right triangle has two legs (indicated as a and b or AC and BC)

Formulas and properties of a right triangle

(see picture above)

a, b- legs of a right triangle

c- hypotenuse

α, β - acute angles of a triangle

S- square

h- the height dropped from the vertex of the right angle to the hypotenuse

m a a from the opposite corner ( α )

m b- median drawn to the side b from the opposite corner ( β )

mc- median drawn to the side c from the opposite corner ( γ )

AT right triangle either leg is less than the hypotenuse(Formula 1 and 2). This property is a consequence pythagorean theorems.

Cosine of any of the acute angles less than one (Formula 3 and 4). This property follows from the previous one. Since any of the legs is less than the hypotenuse, the ratio of the leg to the hypotenuse is always less than one.

The square of the hypotenuse is equal to the sum of the squares of the legs ( Pythagorean theorem). (Formula 5). This property is constantly used in solving problems.

Area of ​​a right triangle equal to half the product of the legs (Formula 6)

Sum of squared medians to the legs is equal to five squares of the median to the hypotenuse and five squares of the hypotenuse divided by four (Formula 7). In addition to the above, there 5 more formulas, so it is recommended that you also read the lesson " Median of a right triangle, which details the properties of the median.

Height of a right triangle is equal to the product of the legs divided by the hypotenuse (Formula 8)

The squares of the legs are inversely proportional to the square of the height dropped to the hypotenuse (Formula 9). This identity is also one of the consequences of the Pythagorean theorem.

Length of the hypotenuse equal to the diameter (two radii) of the circumscribed circle (Formula 10). Hypotenuse of a right triangle is the diameter of the circumscribed circle. This property is often used in problem solving.

Inscribed radius in right triangle circles can be found as half of the expression, which includes the sum of the legs of this triangle minus the length of the hypotenuse. Or as the product of the legs divided by the sum of all sides (perimeter) of a given triangle. (Formula 11)
Sine of an angle opposite this corner leg to hypotenuse(by definition of a sine). (Formula 12). This property is used when solving problems. Knowing the dimensions of the sides, you can find the angle that they form.

Cosine of an angle A (α, alpha) in a right triangle will be equal to relation adjacent this corner leg to hypotenuse(by definition of a sine). (Formula 13)

Triangle - This is one of the most famous geometric shapes. It is used everywhere - not only in the drawings, but also as interior items, details of various designs and buildings. There are several types of this figure - a rectangular one of them. Its distinguishing feature is the presence of a right angle equal to 90°. To find two of the three heights, it is enough to measure the legs. The third is the value between the vertex of the right angle and the midpoint of the hypotenuse. Often in geometry the question is how to find the height of a right triangle. Let's solve this simple problem.

Necessary:

- ruler;
- a book on geometry;
- right triangle.

Instruction:

• Draw a triangle with a right angle ABS, where is the angle ABS equals 90 ° , that is, it is direct. Lower your height H from right angle to hypotenuse AS. The place where the segments touch, mark with a dot D.
• You should get another triangle - adb. Note that it is similar to the existing ABS, since the corners ABS and ADB = 90°, then they are equal to each other, and the angle bad is common to both geometric shapes. By comparing them, we can conclude that the parties AD/AB = BD/BS = AB/AS. From the resulting relations, it can be deduced that AD equals AB2/AS.
• Since the resulting triangle adb has a right angle, while measuring its sides and hypotenuse, you can use the Pythagorean theorem. Here's what it looks like: AB² = AD² + BD². To solve it, use the resulting equality AD. You should get the following: BD² = AB² - (AB²/AC)². Since the measured triangle ABS is rectangular, then BS² equals AS² — AB². Therefore, the side BD2 equals AB²BC²/AC², which with root extraction will be equal to BD=AB*BS/AS.
• Similarly, the solution can be derived using another resulting triangle -
  bds. In this case, it is also similar to the original ABS, thanks to two angles - ABS and BDS = 90°, and the angle DSB is common. Further, as in the previous example, the proportion is displayed in the aspect ratio, where BD/AB = DS/BS = BS/AS. Hence the value D.S. derived through equality BS2/AS. As, AB² = AD*AS , then BS² = DS*AS. Hence we conclude that BD2 = (AB*BS/AS)² or AD*AS*DS*AS/AS², which equals AD*DS. To find the height in this case, it is enough to take the root of the product D.S. and AD.

First of all, a triangle is a geometric figure, which is formed by three points that do not lie on one straight line, which are connected by three segments. To find what the height of a triangle is, it is necessary, first of all, to determine its type. Triangles differ in the size of the angles and the number of equal angles. According to the size of the angles, the triangle can be acute-angled, obtuse-angled and right-angled. According to the number of equal sides, isosceles, equilateral and scalene triangles are distinguished. The height is the perpendicular that is lowered to the opposite side of the triangle from its vertex. How to find the height of a triangle?

How to find the height of an isosceles triangle

An isosceles triangle is characterized by the equality of sides and angles at its base, therefore, the heights of an isosceles triangle drawn to the sides are always equal to each other. Also, the height of this triangle is both a median and a bisector. Accordingly, the height divides the base in half. We consider the resulting right triangle and find the side, that is, the height of the isosceles triangle, using the Pythagorean theorem. Using the following formula, we calculate the height: H \u003d 1/2 * √4 * a 2 - b 2, where: a - the side of this isosceles triangle, b - the base of this isosceles triangle.

How to find the height of an equilateral triangle

A triangle with equal sides is called an equilateral triangle. The height of such a triangle is derived from the formula for the height of an isosceles triangle. It turns out: H = √3/2*a, where a is the side of the given equilateral triangle.

How to find the height of a scalene triangle

A scalene triangle is a triangle in which no two sides are equal to each other. In such a triangle, all three heights will be different. You can calculate the height lengths using the formula: H \u003d sin60 * a \u003d a * (sgrt3) / 2, where a is the side of the triangle, or first calculate the area of ​​​​a particular triangle using the Heron formula, which looks like: S \u003d (p * (p-c) * (p-b)*(p-a))^1/2, where a, b, c are sides of a scalene triangle, and p is its half-perimeter. Each height = 2*area/side

How to find the height of a right triangle

A right triangle has one right angle. The height that passes to one of the legs is at the same time the second leg. Therefore, to find the heights lying on the legs, you need to use the modified Pythagorean formula: a \u003d √ (c 2 - b 2), where a, b are the legs (a is the leg to be found), c is the length of the hypotenuse. In order to find the second height, you need to put the resulting value a in place of b. To find the third height lying inside the triangle, the following formula is used: h \u003d 2s / a, where h is the height of a right-angled triangle, s is its area, a is the length of the side to which the height will be perpendicular.

A triangle is called acute if all its angles are acute. In this case, all three heights are located inside an acute triangle. A triangle is called obtuse if it has one obtuse angle. Two altitudes of an obtuse triangle are outside the triangle and fall on the extension of the sides. The third side is inside the triangle. The height is determined using the same Pythagorean theorem.

General formulas like calculating the height of a triangle

• The formula for finding the height of a triangle through the sides: H= 2/a √p*(p-c)*(p-b)*(p-b), where h is the height to be found, a, b and c are the sides of the given triangle, p is its semi-perimeter, .
• The formula for finding the height of a triangle in terms of angle and side: H=b sin y = c sin ß
• The formula for finding the height of a triangle in terms of area and side: h = 2S / a, where a is the side of the triangle, and h is the height built to side a.
• The formula for finding the height of a triangle in terms of radius and sides: H= bc/2R.