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Proportional segments in a right triangle. Proportional segments in a right triangle Proportional segments in a right triangle 8

Electrodynamics
09.02.2021

Lesson objectives:

Educational:

1.Create conditions for independently deducing relations connecting proportional segments in a right triangle.

  1. Ensure that the acquired knowledge is consolidated when solving problems.

Educational:

1.Ensure the development of independence when performing tasks.

Educational :

1. Foster a culture of communication in a microgroup.

  1. Develop the ability to make decisions and take responsibility for them.

During the classes.

  1. Organizing time.

Guys, listen, how quiet it is!

Lessons started at school.

We won't waste our time

And let's all get to work.

We came here to study

Don't be lazy, but work.

We work diligently

Let's listen carefully.

  1. Lesson motivation.

Dear Guys!

I hope that this lesson will be interesting and of great benefit to everyone. I really want those who are still indifferent to the queen of all sciences to leave our lesson with the deep conviction that geometry is an interesting and necessary subject.

The 19th century French writer Anatole France once remarked: “You can only learn through fun... To digest knowledge, you must absorb it with appetite.”

Let's follow the writer's advice in today's lesson: be active, attentive, and eagerly absorb knowledge that will be useful to you in later life.

3. Updating knowledge. Checking d/z.

Frontal survey:

  1. What is the ratio of two segments called?
  2. In what case do they say that segments AB and CD are proportional to segments A 1 B 1 and C 1 D 1
  3. Define similar triangles
  4. How to read the first sign of similarity of triangles
  5. How to read the second sign of similarity of triangles
  6. How to read the third sign of similarity of triangles
  7. What figures are called similar. What is similarity coefficient?
  8. Right triangle. Legs. Hypotenuse.

Decide No. 570 (orally), 573(1) (written).

  1. Learning new material.

When solving problems, we most often considered acute and obtuse triangles. The elements of a right triangle are related to each other in a slightly different way. Let's look at the drawing.

Properties proportional segments in a right triangle:
1) a leg of a right triangle, there is a proportional mean between the hypotenuse and the projection of this leg onto the hypotenuse;
2) the height of a right triangle drawn from the vertex right angle, there is an average proportional between the projections of the legs onto the hypotenuse.

Historical reference.On the development of practical geometry in ancient Rus'.

Already in the 16th century. the needs of surveying, construction and military affairs led to the creation of handwritten manuals with geometric content. The first work of this kind that has come down to us is called “On laying out the earth, how to lay out the earth.” It is part of the “Book of Soshnogo Letters,” believed to have been written under Ivan IV in 1556. The surviving copy dates back to 1629.

During the dismantling of the Armory Chamber in Moscow in 1775, the instruction “Charter of military, cannon and other matters relating to military military science”, published in 1607 and 1621 and containing some geometric information that boils down to certain techniques for solving problems of finding distances. Here's one example.

To measure the distance from point I to point B (see figure), it is recommended to drive a rod approximately the height of a person into point I. The top of the right angle of the square is attached to the upper end of the rod C so that one of the legs (or its extension) passes through point B. Point 3 of the intersection of the other leg (or its extension) with the ground is marked. Then the distance BYA relates to the length of the rod TsYa in the same way as the length of the rod relates to the distance YAZ. For convenience of calculations and measurements, the rod was divided into 1000 equal parts.

  1. Consolidation of new material.

Decide orally No. 601, in writing No. 610, 600, 604(1), 607(2), 620.

  1. Exercise for the eyes.

Without turning your head, look around the classroom wall around the perimeter clockwise, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and the equal triangle counterclockwise. Turn your head to the left and look at the horizon line, and now at the tip of your nose. Close your eyes, count to 5, open your eyes and...

We'll put our palms to our eyes,
Let's spread our strong legs.
Turning to the right
Let's look around majestically.
And you need to go left too
Look from under your palms.
And - to the right! And further
Over your left shoulder!
Now let's continue working.

  1. Independent work.

Work in pairs: solve No. 604(2) (written)

8. Lesson summary. Reflection.

  • What do you remember most about the lesson?
  • What surprised you?
  • What did you like the most?
  • What do you want the next lesson to look like?

Homework: learn paragraph 14, solve No. 604(3), 607(3), 573(2).

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Slide captions:

Proportional segments in a right triangle Geometry grade 8

Homework

1. Problem 3, 5 A B C N M 3 4 Given: MN || A.C. Find: Р∆АВС

A B C D M N P Q MNPQ is a parallelogram? 2. Problem

Similarity of right triangles A B C A 1 B 1 C 1 If an acute angle of one right triangle is equal to an acute angle of another right triangle, then such right triangles are similar

Proportional mean A B C D X Y The segment XY is called the proportional mean (geometric mean) for segments AB and CD if

Solve the problems: 1. Is a segment of length 8 cm the average proportional between segments with lengths of 16 cm and 4 cm? 2. Is a segment of length 9 cm the average proportional between segments with lengths of 15 cm and 6 cm? 3. Is a segment of length cm the average proportional between segments with lengths 5 cm and 4 cm? yes no yes

Proportional segments in a right triangle A B C H The height of a right triangle drawn from the vertex of a right angle is the average proportional to the segments into which the hypotenuse is divided by this height

Proportional segments in a right triangle A B C H 9 4? Task 1.

Proportional segments in a right triangle A B C H 9 7? Task 2.

Proportional segments in a right triangle A B C N A leg of a right triangle is the mean proportional to the hypotenuse and the projection of this leg onto the hypotenuse.

Proportional segments in a right triangle A B C H 21 4? Task 3.

A B C N 20 30 ? Task 4.

Homework

Solve problem 5 2 ? ? ? Solve problem 9 4 ? ? ? Solve triangle

A B C N 20 15 ? Task. In a triangle whose sides are 15, 20 and 25, the altitude is drawn to its longer side. Find the segments into which the height divides this side 25

A B C N 20 15 ? Task 5. In a triangle whose sides are 15, 20 and 25, the altitude is drawn to its longer side. Find the segments into which the height divides this side 25

Sections: Mathematics

Class: 8

Type of lesson: combined.

Didactic goal: creating conditions for awareness and comprehension of the concept of “proportional average”, improving the skills of finding proportional segments based on the similarity of triangles, checking the level of assimilation of knowledge and skills on the topic.

Tasks:

  • establish a correspondence between the sides of a right triangle, the height drawn to the hypotenuse and the segments of the hypotenuse;
  • introduce the concept of average proportional;
  • develop the ability to apply acquired knowledge to solve practical problems;

Educational materials: textbook “Geometry 7-9” by L. S. Atanasyan, presentation “Proportional segments in a right triangle.” Annex 1 .

Expected results:

Personal

  • The ability to determine the boundary between knowledge and ignorance.
  • Ability to express thoughts mathematically correctly.
  • Ability to recognize incorrect statements.

Metasubject

  • The ability to plan your activities to solve a learning problem.
  • The ability to build a chain of logical reasoning.
  • The ability to give a verbal formulation to a fact written in the form of a formula.

Subject

  • The ability to find similar triangles and prove their similarity.
  • The ability to express the legs of a right triangle and the height drawn from the vertex of a right angle through segments of the hypotenuse.
  • Ability to read mathematical notation using the concept of “proportional average.”

Lesson outline plan.

1. Organizational moment. Organization of attention; volitional self-regulation. (Each student is given worksheets for the lesson for two options). Appendix 2 ,Appendix 3 .

2. Repetition: Let's repeat the basic information of the topic “Similar triangles” Slide 1

  • Define similar triangles
  • How to read the first sign of similarity of triangles
  • How to read the second sign of similarity of triangles
  • How to read the third sign of similarity of triangles
  • What is similarity coefficient?
  • Right triangle. Legs. Hypotenuse.

A test to determine the truth or falsity of statements (answer “yes” or “no”). Slide 2

  • Two triangles are similar if their angles are respectively equal and their similar sides are proportional.
  • Two equilateral triangles are always similar.
  • If three sides of one triangle are respectively proportional to three sides of another triangle, then such triangles are similar.
  • The sides of one triangle have lengths of 3, 4, 6 cm, the sides of the other triangle are 9, 14, 18 cm. Are these triangles similar?
  • The perimeters of similar triangles are equal.
  • If two angles of one triangle are 60° and 50°, and two angles of another triangle are 50° and 80°, then the triangles are similar.
  • Two right triangles are similar if they have equal acute angles.
  • Two isosceles triangles are similar.
  • If two angles of one triangle are respectively equal to two angles of another triangle, then such triangles are similar.
  • If two sides of one triangle are respectively proportional to two sides of another triangle, then the triangles are similar.

Key to the test: 1. yes; 2. yes; 3. yes; 4. no; 5. no; 6. no; 7. yes; 8. no; 9. yes; 10. no.

The test verification form is mutual verification. Answers and verification are carried out in the worksheet for the lesson.

3. Theoretical task in groups. The class is divided into three groups. Each group receives a task. Appendix 4 .

Group No. 1

  1. Prove the similarity of the “left” and “right” right triangles.
  2. Write down the proportionality of the legs.
  3. Express the height from the proportion.

Group No. 2

According to a pre-prepared drawing of a right triangle (Figure 1)

  1. Prove the similarity of the “left” and “large” right triangles.
  2. Express from the proportion BC.

Group No. 3

According to a pre-prepared drawing of a right triangle (Figure 1)

  1. Prove the similarity of the “right” and “large” right triangles.
  2. Write down the proportionality of similar sides.
  3. Express from the proportion AC.

Write down the proof of these statements on the board using pre-made drawings and in notebooks. One person from the group is called to the board.

4. Formulation of the lesson topic. In all three tasks, we made some relationships. What can you call the elements included in these relationships? Answer: proportional segments. Let's clarify the proportional segments in...? Answer: in a right triangle. So, guys, the topic of our lesson? Answer: “Proportional segments in a right triangle.” Slide 3

5. Formulation of proven statements

Before working further, let's introduce some new concepts and notations.
What is the arithmetic mean of two numbers?
Answer: The arithmetic mean of the numbers m and n is the number a equal to half the sum of the numbers m and n
Write down the formula for the arithmetic mean of the numbers m and n.
Let us formulate the definition of the geometric mean of two numbers: the number a is called the geometric mean (or proportional mean) for the numbers m and n if the equality is satisfied Slide 4
Let's solve several exercises to consolidate these definitions. Slide 5
1. Find the arithmetic mean and geometric mean of the numbers 3 and 12.
2. Find the length of the average proportional (geometric average) segments MN and KP, if MN = 9 cm, KP = 27 cm
Let us introduce the concept of projection of a leg onto the hypotenuse. Slide 6.
Now, using new concepts, we will try to formulate the conclusions proven during group work.
Using this slide, try to formulate a statement that was proven by the second and third groups. Slide 7
Write down this statement using the new notation (projection of a leg onto the hypotenuse) and then formulate it using the definition of projection of a leg onto the hypotenuse. Slide 8
Based on this slide, try to formulate a statement that was proven by students in the third group. Slide 9
Write down this statement using the new notation (projection of a leg onto the hypotenuse) and then formulate it using the definition of projection of a leg onto the hypotenuse. Slide 10

6. Blitz survey to consolidate the studied formulas. Slide 11-12

  • In a right triangle ABC, the altitude CD is drawn from the vertex of right angle C. AD = 16, DB = 9. Find AC, AB, CB and CD. Slide 11
  • In a right triangle ABC, the altitude CD is drawn from the vertex of right angle C. AD = 18, DB = 2. Find AC, AB, CB and CD. Slide 12
  • In a right triangle ABC, the altitude CH is drawn from the vertex of right angle C. CA = 6, AN = 2. Find NV. Slide 13

Test to check initial mastery of material

In the presentation, open the slide with the derived formulas (Slide 14). The worksheets have a test printed on them: complete the test by writing the correct answers on the chart. Then mutual checking (Slide 15) using ready-made answers in the presentation.

Homework

Each student is given a memo with formulas and the text of homework problems with tips (a plan for the step-by-step completion of each task) Appendix 5 .

9. Reflection

Summarize the lesson. Collect worksheets and grade each student's lesson.

Literature.

  1. http://gorkunova.ucoz.ru/ Handouts for the workshop on the topic "Proportional segments in a right triangle"
  2. Presentation “Proportional segments in a right triangle” Savchenko E.M. Polyarnye Zori, Murmansk region.