Rectangular

triangles

Geometry, 7th grade

To the textbook by L.S. Atanasyan

mathematics teacher of the highest category

Municipal educational institution "Upshinskaya basic secondary school"

Orsha district of the Republic of Mari El

AC, BC – legs

AB - hypotenuse

Property 1 0 . The sum of the acute angles of a right triangle is 90 0.

Task 1. Find angle A of right triangle ABC with right angle C if: a) ے B = 32 0; b) ے B is 2 times less than angle A; c) ے B is 20 0 less than angle A.

Task 2.

Task 3.

Angle A:

BC – leg lying opposite angle A

AC – leg adjacent to angle A

Angle B:

AC - leg, ...

BC - leg, ...

Name the legs opposite angles N and K

Name the legs adjacent to angles N and K

0

Task. Prove that 0 , is equal to half the hypotenuse.

Property 2 0 . A leg of a right triangle lying opposite an angle of 30 0 , is equal to half the hypotenuse.

Right triangle with an angle of 30 0

Task. Prove that 0 .

Property 3 0 . If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is 30 0 .

Right triangle with angle 30 0

Problem 4 .

AB = 12 cm. Find BC

Task 5.

BC = 7.5 cm. Find AB

Task 6.

AB + BC = 15 cm.

Find AB and BC

Right triangle with angle 30 0

Task 7.

AC = 4 cm. Find AB

Task 8.

AB - AC = 15 cm.

Find AB and AC

Right triangle with angle 30 0

Problem 9 .

Find the acute angles of right triangle ABC if AB = 12 cm, BC = 6 cm.

Right triangle with angle 30 0

Problem 10 .

Find the acute angles of a right triangle if the angle between the bisector and the altitude drawn from the vertex right angle, is equal to 15 0.

SC - bisector

CM - height

Right triangle with angle 30 0

Problem 11 .

In an isosceles triangle, one of the angles is 120 0 and the base is 4 cm. Find the height drawn to the side.

AM - height

Right triangle with angle 30 0

Problem 12 .

The altitude drawn to the lateral side of an isosceles triangle bisects the angle between the base and the bisector. Find the angles of an isosceles triangle.

AK – bisector of angle A

AM - height

Right triangle with angle 30 0

Problem 14 .

Prove that if the triangle is right-angled, then the median drawn from the vertex of the right angle is equal to half the hypotenuse.

Property 4 0 .

ΔАВС - rectangular

SM – median

We got a contradiction!

Right triangle with angle 30 0

Problem 13 .

Prove that if the median of a triangle is equal to half the side to which it is drawn, then the triangle is right-angled.

VM – median

Prove: ΔABC - rectangular

Property 5 0 .

Some properties of right triangles

Property 1 0 . The sum of the acute angles of a right triangle is 90 0 .

Property 2 0 . A leg of a right triangle lying opposite an angle of 30 0 , equal to half the hypotenuse .

Property 3 0 . If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is 30 0 .

Property 4 0 . In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse.

Property 5 0 . If the median of a triangle is equal to half the side to which it is drawn, then this triangle is right-angled.

Definition.Right triangle - a triangle, one of whose angles is right (equal to ).

A right triangle is a special case of an ordinary triangle. Therefore, all the properties of ordinary triangles for right triangles are preserved. But there are also some particular properties due to the presence of a right angle.

Common designations (Fig. 1):

- right angle;

- hypotenuse;

- legs;

Rice. 1.

WITHproperties of a right triangle.

Property 1. The sum of the angles and a right triangle is equal to .

Proof. Recall that the sum of the angles of any triangle is equal to . Taking into account the fact that , we find that the sum of the remaining two angles is equal to That is,

Property 2. In a right triangle hypotenuse more than any of legs(is the largest side).

Proof. Recall that in a triangle, the larger side lies opposite the larger angle (and vice versa). From Property 1 proven above it follows that the sum of the angles and a right triangle is equal to . Since the angle of a triangle cannot be equal to 0, then each of them is less than . This means that it is the largest, which means that the largest side of the triangle lies opposite it. This means that the hypotenuse is the longest side of a right triangle, that is: .

Property 3. In a right triangle, the hypotenuse is less than the sum of the legs.

Proof. This property becomes obvious if we recall triangle inequality.

Triangle inequality

In any triangle, the sum of any two sides is greater than the third side.

Property 3 immediately follows from this inequality.

Note: despite the fact that each of the legs individually is smaller than the hypotenuse, their sum turns out to be greater. In a numerical example it looks like this: , but .

V:

1st sign (on 2 sides and the angle between them): If triangles have two equal sides and the angle between them, then such triangles are congruent.

2nd sign (by side and two adjacent angles): if triangles have equal sides and two angles adjacent to a given side, then such triangles are congruent. Note: Using the fact that the sum of the angles of a triangle is constant and equal to , it is easy to prove that the condition of “adherence” of the angles is not necessary, that is, the sign will be true in the following formulation: “... the side and two angles are equal, then...”.

3rd sign (on 3 sides): If triangles have all three sides equal, then such triangles are congruent.

Naturally, all these signs remain true for right triangles. However, right triangles have one significant feature - they always have a pair of equal right angles. Therefore, these signs are simplified for them. So, let’s formulate the signs of equality of right triangles:

1st sign (on two sides): if right triangles have pairwise equal legs, then such triangles are equal to each other (Fig. 2).

Given:

Rice. 2. Illustration of the first sign of equality of right triangles

Prove:

Proof: in right triangles: . This means that we can use the first sign of equality of triangles (by 2 sides and the angle between them) and get: .

2-th sign (by leg and angle): if the leg and acute angle of one right triangle are equal to the leg and acute angle of another right triangle, then such triangles are congruent (Fig. 3).

Given:

Rice. 3. Illustration of the second sign of equality of right triangles

Prove:

Proof: Let us immediately note that the fact that the angles adjacent to equal legs are equal is not fundamental. Indeed, the sum of the acute angles of a right triangle (by property 1) is equal to . This means that if one pair of these angles is equal, then the other is equal (since their sums are the same).

The proof of this characteristic comes down to using second sign of equality of triangles(on 2 corners and one side). Indeed, by condition, the legs and a pair of adjacent angles are equal. But the second pair of adjacent angles consists of angles . This means that we can use the second criterion for the equality of triangles and get: .

3rd sign (by hypotenuse and angle): if the hypotenuse and acute angle of one right triangle are equal to the hypotenuse and acute angle of another right triangle, then such triangles are congruent (Fig. 4).

Given:

Rice. 4. Illustration of the third sign of equality of right triangles

Prove:

Proof: to prove this sign you can immediately use second sign of equality of triangles- on a side and two angles (more precisely, a corollary, which states that the angles do not have to be adjacent to the side). Indeed, according to the condition: , , and from the properties of right triangles it follows that . This means that we can use the second criterion for the equality of triangles and get: .

4th sign (by hypotenuse and leg): if the hypotenuse and leg of one right triangle are equal, respectively, to the hypotenuse and leg of another right triangle, then such triangles are equal to each other (Fig. 5).

Given:

Rice. 5. Illustration of the fourth sign of equality of right triangles

Prove:

Proof: To prove this criterion, we will use the criterion for the equality of triangles, which we formulated and proved in the last lesson, namely: if triangles have two equal sides and a larger angle, then such triangles are equal. Indeed, by condition we have two equal sides. In addition, according to the property of right triangles: . It remains to prove that the right angle is the largest in the triangle. Let's assume that this is not the case, which means there must be at least one more angle that is greater than . But then the sum of the angles of the triangle will already be greater. But this is impossible, which means that such an angle cannot exist in a triangle. This means that the right angle is the largest in a right triangle. This means that you can use the feature formulated above and get: .

Let us now formulate one more property that is characteristic only of right triangles.

Property

The leg lying opposite the angle in is 2 times smaller than the hypotenuse(Fig. 6).

Given:

Rice. 6.

Prove:AB

Proof: Let's perform an additional construction: extend the straight line beyond the point to a segment equal to . Let's get a point. Since the angles and are adjacent, their sum is equal to . Since , then the angle .

This means that right triangles (on two sides: - general, - by construction) are the first sign of equality of right triangles.

From the equality of triangles it follows that all corresponding elements are equal. Means, . Where: . In addition, (from the equality of the same triangles). This means that the triangle is isosceles (since its base angles are equal), but an isosceles triangle, one of the angles of which is equal to , is equilateral. It follows from this, in particular, that .

Property of a leg lying opposite an angle in

It is worth noting that the opposite statement is also true: if in a right triangle the hypotenuse is twice the size of one of the legs, then the acute angle opposite this leg is equal to .

Note: sign means that if any statement is true, then the triangle is right-angled. That is, the feature allows you to identify a right triangle.

It is important not to confuse a sign with property- that is, if the triangle is right-angled, then it has the following properties... Often the signs and properties are mutually inverse, but not always. For example, the property of an equilateral triangle: an equilateral triangle has an angle. But this will not be a sign of an equilateral triangle, since not every triangle that has an angle, is equilateral.

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:

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:

Sinus acute angle equal to the ratio of the opposite side to the hypotenuse

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

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

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

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

It's very convenient!

Signs of equality of right triangles

I. On two sides

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:

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

It is necessary that in both triangles the leg was adjacent, or in both it was opposite.

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

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:

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..., 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(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:

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

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?

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: