Formulas connecting proportional line segments in a right-angled triangle. Proportional line segments in a right triangle

Similarity mark for right triangles

Let us first introduce a similarity criterion for right-angled triangles.

Theorem 1

Similarity mark for right triangles: two right-angled triangles are similar when they have one equal acute angle (fig. 1).

Figure 1. Similar right-angled triangles

Evidence.

Let us be given that $ \\ angle B \u003d \\ angle B_1 $. Since triangles are rectangular, then $ \\ angle A \u003d \\ angle A_1 \u003d (90) ^ 0 $. Therefore, they are similar in the first sign of similarity of triangles.

The theorem is proved.

Height theorem in a right triangle

Theorem 2

The height of a right-angled triangle, drawn from the apex of the right angle, divides the triangle into two similar right-angled triangles, each of which is similar to this triangle.

Evidence.

Let us be given a right-angled triangle $ ABC $ with a right angle $ C $. Let's draw the height $ CD $ (Fig. 2).

Figure 2. Illustration of Theorem 2

Let us prove that the triangles $ ACD $ and $ BCD $ are similar to the triangle $ ABC $ and that the triangles $ ACD $ and $ BCD $ are similar to each other.

Since $ \\ angle ADC \u003d (90) ^ 0 $, the triangle $ ACD $ is rectangular. The triangles $ ACD $ and $ ABC $ have a common angle $ A $, therefore, by Theorem 1, the triangles $ ACD $ and $ ABC $ are similar.

Since $ \\ angle BDC \u003d (90) ^ 0 $, the triangle $ BCD $ is rectangular. The triangles $ BCD $ and $ ABC $ have a common angle $ B $, therefore, by Theorem 1, the triangles $ BCD $ and $ ABC $ are similar.

Consider now the triangles $ ACD $ and $ BCD $

\\ [\\ angle A \u003d (90) ^ 0- \\ angle ACD \\] \\ [\\ angle BCD \u003d (90) ^ 0- \\ angle ACD \u003d \\ angle A \\]

Therefore, by Theorem 1, the triangles $ ACD $ and $ BCD $ are similar.

The theorem is proved.

Proportional mean

Theorem 3

The height of a right-angled triangle, drawn from the vertex of the right angle, is the proportional average for the segments into which the height divides the hypotenuse of this triangle.

Evidence.

By Theorem 2, we have that the triangles $ ACD $ and $ BCD $ are similar, hence

The theorem is proved.

Theorem 4

The leg of a right-angled triangle is the average proportional between the hypotenuse and the segment of the hypotenuse, enclosed between the leg and the height drawn from the apex of the angle.

Evidence.

In the proof of the theorem, we will use the notation from Figure 2.

By Theorem 2, we have that the triangles $ ACD $ and $ ABC $ are similar, hence

The theorem is proved.

Lesson objectives:

1. introduce the concept of a proportional average (geometric mean) of two segments;
2. consider the problem of proportional segments in a right-angled triangle: the property of the height of a right-angled triangle, drawn from the vertex of a right angle;
3. to form students' skills in using the studied topic in the process of solving problems.

Lesson type: lesson in learning new material.

Plan:

1. Organizational moment.
2. Knowledge update.
3. Studying the property of the height of a right-angled triangle, drawn from the vertex of a right angle:
   - preparatory stage;
   - introduction;
   - assimilation.
4. Introduction of the concept of a mean proportional to two segments.
5. Mastering the concept of the average proportional to two segments.
6. Proof of the consequences:
   - the height of a right-angled triangle, drawn from the top of the right angle, is the average proportional between the segments into which the hypotenuse is divided by this height;
   - the leg of a right-angled triangle is the average proportional between the hypotenuse and the segment of the hypotenuse, enclosed between the leg and the height.
7. Solving problems.
8. Summarizing.
9. Homework setting.

During the classes

I. ORGMOMENT

- Hello guys, have a seat. Is everyone ready for the lesson?

Getting started.

II. KNOWLEDGE UPDATE

- What important mathematical concept did you meet in the previous lessons? ( with the concept of similarity of triangles)

- Let's remember which two triangles are called similar? (two triangles are said to be similar if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other triangle)

- What do we use to prove the similarity of two triangles? (

- Formulate these signs (formulate three criteria for the similarity of triangles)

III. STUDY OF THE HEIGHT PROPERTIES OF A RECTANGULAR TRIANGLE DRAWED FROM THE TOP OF A RIGHT ANGLE

a) preparatory stage

- Guys, please look at the first slide. ( application) Here are two right-angled triangles - and. and - heights and, respectively. .

Task 1.a) Determine if and are similar.

- What do we use to prove the similarity of triangles? ( signs of similarity of triangles)

(the first sign, since nothing is known about the sides of the triangles in the problem)

... (Two pairs: 1.∟B \u003d ∟B1 (straight lines), 2. 2A \u003d ∟A 1)

- Make a conclusion. ( by the first sign of similarity of triangles ~)

Task 1.b) Determine if and are similar.

- What sign of similarity will we use and why? (the first sign, since in the problem nothing is known about the sides of the triangles)

- How many pairs of equal angles do we need to find? Find these pairs (since triangles are rectangular, one pair of equal angles is enough: ∟A \u003d ∟A 1)

- Make a conclusion. (by the first sign of similarity of triangles, we conclude that these triangles are similar).

As a result of the conversation, slide 1 looks like this:

b) discovery of the theorem

Task 2.

- Determine if and, and are similar. As a result of the conversation, the answers are built, which are reflected on the slide.

- The picture indicated that. Did we use this degree measure when answering the questions of the tasks? ( No, did not use)

- Guys, draw a conclusion: into which triangles does the right-angled triangle divide the height drawn from the vertex of the right angle? (conclude)

- The question arises: will these two right-angled triangles, into which the height breaks the right-angled triangle, be similar to each other? Let's try to find pairs of equal angles.

As a result of the conversation, a record is built:

- Now let's make a full conclusion. ( CONCLUSION: the height of a right-angled triangle, drawn from the vertex of a right angle, divides the triangle into two similar

- So. we have formulated and proved the theorem on the property of the height of a right-angled triangle.

Let us establish the structure of the theorem and draw a drawing. What is given in the theorem and what needs to be proved? Students write in a notebook:

- Let us prove the first item of the theorem for the new drawing. What sign of similarity will we use and why? (The first, because in the theorem nothing is known about the sides of triangles)

- How many pairs of equal angles do we need to find? Find these pairs. (In this case, one pair is sufficient: ∟A-common)

- Make a conclusion. The triangles are similar. As a result, a sample of the theorem is shown

- Write down the second and third points at home yourself.

c) assimilation of the theorem

- So, formulate the theorem again (The height of a right-angled triangle, drawn from the vertex of the right angle, divides the triangle into two similarright-angled triangles, each of which is similar to this one)

- How many pairs of similar triangles in the construction "in a right-angled triangle the height is drawn from the vertex of the right angle" does this theorem allow to find? ( Three pairs)

Students are offered the following task:

IV. INTRODUCTION OF THE CONCEPT OF THE AVERAGE PROPORTIONAL OF TWO LEGS

- And now we will study a new concept with you.

Attention!

Definition. Section XY called average proportional (geometric mean) between segments AB and CD, if a

(write in a notebook).

V. ASSIGNMENT OF THE CONCEPT OF THE AVERAGE PROPORTIONAL OF TWO INTERSECTS

- Now let's turn to the next slide.

Exercise 1.Find the length of the mean of the proportional segments MN and KP, if MN \u003d 9 cm, KP \u003d 16 cm.

- What is given in the task? ( Two segments and their lengths: MN \u003d 9 cm, KP \u003d 16 cm)

- What do you need to find? ( The length of the average proportional to these segments)

- What is the formula for the proportional mean and how do we find it?

(We substitute the data into the formula and find the length of the average prop.)

Task number 2.Find the length of the segment AB, if the average proportional to the segments AB and CD is 90 cm and CD \u003d 100 cm

- What is given in the task? (the length of the segment CD \u003d 100 cm and the average proportional to the segments AB and CD is 90 cm)

- What do you need to find in the problem? ( Segment length AB)

- How are we going to solve the problem? (We write the formula for the average of the proportional segments AB and CD, express the length AB from it and substitute the problem data.)

Vi. CONCLUSION OF CONSEQUENCES

- Well done boys. Now let's return to the similarity of triangles, which we proved in the theorem. Formulate the theorem again. ( The height of a right-angled triangle, drawn from the vertex of the right angle, divides the triangle into two similarright-angled triangles, each of which is similar to a given)

- Let's first use the similarity of triangles and. What follows from this? ( By definition of similarity, sides are proportional to similarities)

- What equality will be obtained using the main property of proportion? ()

- Express CD and draw a conclusion (;.

Conclusion: the height of a right-angled triangle, drawn from the vertex of a right angle, is the proportional average between the segments into which the hypotenuse is divided by this height)

- And now prove yourself that the leg of a right-angled triangle is the average proportional between the hypotenuse and the segment of the hypotenuse enclosed between the leg and the height. Let's find from - ... the segments into which the hypotenuse is divided by this height )

The leg of a right-angled triangle is the average proportional between ... (- ... the hypotenuse and the segment of the hypotenuse enclosed between this leg and the height )

- Where do we apply the learned statements? ( When solving problems)

IX. HOME ASSIGNMENT

d / s: No. 571, No. 572 (a, e), independent work in a notebook, theory.

Today we invite your attention to another presentation on an amazing and mysterious subject - geometry. In this presentation we will introduce you to a new property of geometric shapes, in particular, the concept of proportional line segments in right-angled triangles.

First, you need to remember what a triangle is? This is the simplest polygon with three vertices connected by three line segments. A rectangular triangle is called a triangle in which one of the angles is 90 degrees. You have already familiarized yourself with them in more detail in our previous training materials presented to your attention.

So, returning to our topic today, let us denote in order that the height of a right-angled triangle, drawn from an angle of 90 degrees, divides it into two triangles, which are similar both to each other and to the original. All the figures and graphs you are interested in are given in the proposed presentation, and we recommend that you refer to them, accompanying the described explanation.

A graphic example of the above thesis can be seen on the second slide. Based on the first sign of the similarity of triangles, triangles are similar, since they have two identical angles. If you specify in more detail, then the height lowered to the hypotenuse forms a right angle with it, that is, there are already the same angles, and each of the formed angles also has one common angle as the initial one. The result is two angles that are equal to each other. That is, the triangles are similar.

Let us also denote what does the concept of “proportional mean” or “geometric mean” mean? This is a certain XY segment for segments AB and CD, when it is equal to the square root of the product of their lengths.

From which it also follows that the leg of a right-angled triangle is the geometric mean between the hypotenuse and the projection of this leg onto the hypotenuse, that is, the other leg.

Another of the properties of a right triangle is that its height, drawn from an angle of 90 °, is the average proportional between the projections of the legs on the hypotenuse. If you refer to the presentation and other materials offered to your attention, you will see that there is a proof of the indicated thesis in a very simple and accessible form. We have already proved that the resulting triangles are similar to each other and to the original triangle. Then, using the ratio of the legs of these geometric figures, we come to the fact that the height of a right-angled triangle is directly proportional to the square root of the product of the segments that were formed as a result of lowering the height from the right angle of the original triangle.

The latter in the presentation indicated that the leg of a right-angled triangle is the geometric mean for the hypotenuse and its segment located between the leg and the height, drawn from an angle of 90 degrees. This case should be considered from the side that the indicated triangles are similar to each other, and the leg of one of them is obtained by the hypotenuse of the other. But you will get acquainted with this in more detail by studying the proposed materials.

Similarity mark for right triangles

Let us first introduce a similarity criterion for right-angled triangles.

Theorem 1

Similarity mark for right triangles: two right-angled triangles are similar when they have one equal acute angle (fig. 1).

Figure 1. Similar right-angled triangles

Evidence.

Let us be given that $ \\ angle B \u003d \\ angle B_1 $. Since triangles are rectangular, then $ \\ angle A \u003d \\ angle A_1 \u003d (90) ^ 0 $. Therefore, they are similar in the first sign of similarity of triangles.

The theorem is proved.

Height theorem in a right triangle

Theorem 2

The height of a right-angled triangle, drawn from the apex of the right angle, divides the triangle into two similar right-angled triangles, each of which is similar to this triangle.

Evidence.

Let us be given a right-angled triangle $ ABC $ with a right angle $ C $. Let's draw the height $ CD $ (Fig. 2).

Figure 2. Illustration of Theorem 2

Let us prove that the triangles $ ACD $ and $ BCD $ are similar to the triangle $ ABC $ and that the triangles $ ACD $ and $ BCD $ are similar to each other.

Since $ \\ angle ADC \u003d (90) ^ 0 $, the triangle $ ACD $ is rectangular. The triangles $ ACD $ and $ ABC $ have a common angle $ A $, therefore, by Theorem 1, the triangles $ ACD $ and $ ABC $ are similar.

Since $ \\ angle BDC \u003d (90) ^ 0 $, the triangle $ BCD $ is rectangular. The triangles $ BCD $ and $ ABC $ have a common angle $ B $, therefore, by Theorem 1, the triangles $ BCD $ and $ ABC $ are similar.

Consider now the triangles $ ACD $ and $ BCD $

\\ [\\ angle A \u003d (90) ^ 0- \\ angle ACD \\] \\ [\\ angle BCD \u003d (90) ^ 0- \\ angle ACD \u003d \\ angle A \\]

Therefore, by Theorem 1, the triangles $ ACD $ and $ BCD $ are similar.

The theorem is proved.

Proportional mean

Theorem 3

The height of a right-angled triangle, drawn from the vertex of the right angle, is the proportional average for the segments into which the height divides the hypotenuse of this triangle.

Evidence.

By Theorem 2, we have that the triangles $ ACD $ and $ BCD $ are similar, hence

The theorem is proved.

Theorem 4

The leg of a right-angled triangle is the average proportional between the hypotenuse and the segment of the hypotenuse, enclosed between the leg and the height drawn from the apex of the angle.

Evidence.

In the proof of the theorem, we will use the notation from Figure 2.

By Theorem 2, we have that the triangles $ ACD $ and $ ABC $ are similar, hence

The theorem is proved.

Lesson 40. Proportional line segments in a right-angled triangle. C. b. a. h. C. bc. H. ac. A. B. The height of a right-angled triangle, drawn from the apex of the right angle, divides the triangle into 2 similar right-angled triangles, each of which is similar to this triangle. A similarity mark for right-angled triangles. Two right-angled triangles are similar if they have an equal acute angle. The XY segment is called the proportional average (geometric mean) for the AB and CD segments, if Property 1. The height of a right-angled triangle, drawn from the vertex of the right angle, is the proportional average between the projections of the legs to the hypotenuse. Property 2. The leg of a right-angled triangle is the average proportional between the hypotenuse and the projection of this leg onto the hypotenuse.

Geometry grade 8

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