Presentation on the topic similar triangles. Similarity of triangles. The first sign of similarity is presentation. Practical applications of triangle similarity
Slide 2. This slide shows how the Pythagorean Theorem is presented in the textbook. Text and finished drawing. In a presentation, we can “revive” a static drawing from a textbook, i.e. show the successive steps of construction, show the dynamics of additional constructions necessary for the proof.
I work in a classroom with a remote mouse so I can control the presentation and work one-on-one with students at the same time. I consider this the main advantage of using presentations in a geometry lesson. I am not “tied” to the board or computer; I have extra time for individual work. Appeared free time allows me to go around all the children and check the correctness of the drawing in the notebooks. It sometimes feels like there are two teachers in the class. The first one works “in real life” individually–
It's me. The second virtual teacher shows the construction steps - this is a computer. I have the opportunity, at the request of the children, to repeat the construction steps and scroll the mouse wheel back.
Slide 3. Pythagorean theorem. Algorithm for working with the module in a lesson.
- To prove it, we need to complete the triangle to a square. The teacher demonstrates the construction on a slide, working with a remote mouse, and leads individual work with students.
-To prove it, we calculate the area of the constructed square in two ways.
How can you calculate the area of a square? Frontal work on the idea of proof.
First way. S = a². The side of the square is (a+b), then S = (a+b)².
The second method of calculation is using the property of areas: the area of a square is equal to the sum of the areas of four right triangles and the area of a square with side c.
Let us equate the right-hand sides of these equalities. I call a student to the board. We draw up the transformations with chalk on a blackboard.
Slide 4. A technically more complex slide. Animations were used: rotations, paths of movement. This module uses an animated character to accompany the explanation.
Slide 5. Using a presentation, you can provide a significantly larger amount of information in the lesson. For example, imagine other ways to prove the theorem.
And how many problems can be offered to test the proven theorems! For example, here are the problems I compiled to practice writing down the formulation of the Pythagorean theorem.
Slides 6, 7 for oral work. Technically, these modules are quite simple. Algorithm of work in the lesson.
Students must formulate the property of the diagonals of a rhombus and name all the triangles. And then for each triangle write down the Pythagorean theorem.
By making minor changes to the slides, these tasks can be offered in the next lesson as tasks with subsequent testing.
Algorithm for organizing work in the classroom. Slides 8, 9.
Slide 8. Mathematical dictation. Write sequentially the Pythagorean theorem for each triangle. Triangles appear when you click on any part of the slide (but not on the curtain). Let's move on to slide 9. For four more triangles we write down the theorem. Click the button to return back to slide 8. Click on the curtain to open the answers. Self-check or mutual check. Go to slide 9, click on the curtain to open the answers. During the lesson, you can schedule 1 or more slides with independent work followed by a self-test.
Slide 10. Algorithms for organizing work on a theorem in a lesson may be different. In one class we will work with the theorem in one way, in another class we will organize the work differently. For example. I will look at the property of the angles of an isosceles triangle.
1 way to organize work on the theorem.
Teacher. We highlight the condition and conclusion of the theorem.
Students formulate what is “given” in the theorem and what needs to be “proved.”
Teacher. Please complete my prompt sentences. The equality of angles usually follows from... Students continue... from the equality of triangles.
Teacher. So we need triangles. To make the triangles appear, we will make an additional construction. Figure out how to split a triangle into two equal triangles? Let's construct the bisector ВD. (I stop the presentation at this point.)
Students usually immediately see congruent triangles. Let us prove the equality of the triangles. One student is invited to the blackboard and writes down the proof of the equality of triangles with chalk on the board. Writes out equal elements. Draws a conclusion about the equality of the triangles and names the sign. The final conclusion is that the angles at the base are equal.
Teacher. Let's check and repeat the proof. (Continues showing the presentation).
Thus, the student completes the proof independently, and the teacher shows it again through the projector, and a step-by-step analysis of the proof occurs.
2 way to work on the theorem.
If there are no students in the class who can prove the theorem on their own and make competent sequential notes on the steps of the proof from beginning to end.
We review the entire course of the proof from beginning to end. We make a drawing, formulate the conditions and conclusion of the theorem. We draw up a drawing in a notebook, given, prove it.
Let's discuss the proof frontally. Together we look for equal elements of the triangles that appear in the drawing. After an oral analysis of the theorem, we call a student to the board who can reconstruct the proof. So we formulate the task “Restore the proof” for him. Use the wheel on the mouse to return to the beginning of the proof (Given, prove, DP is a bisector).
So, in the first case, students prove the theorem on their own
. After that, we show the proof through the projector and generalize. In the second case, we first view the proof through the projector, and then ask restore the evidence
.
But there are theorems that students cannot prove on their own. Here the computer will come to the aid of the teacher. In the presentation, you can “revive” the drawing, animate the successive steps of the proof, using color highlighting of the figures, and make the proof more understandable.
Slides 11 – 13.
Slide 11 provides a visual cue from the computer - the words “If” and “then” are highlighted in red. It is not difficult to formulate the conditions and conclusion of the theorem.
On slide 12 is an animated proof. In a prepared class, you can first review the theorem and then have them reconstruct the proof with chalk on the board. After viewing the proof, you can right-click to select Screen - Black screen.
In another class, you can draw up the proof in a notebook at the same time as showing it. The slide shows the notes that should be written in the notebook.
You can also give two more cases, which we will offer for independent proof (for example, do it at home if you wish). After completing the entries in the notebook, we review the evidence again. The teacher repeats all steps.
I also used the same algorithm. For example, simultaneously with the demonstration, students wrote down the proof in their notebooks. Those. We look at it at the same time, discuss it frontally, and write down the proof in our notebooks. After completing this work, I use the mouse wheel to return to the beginning of the theorem. I invite the student to the screen. With a pointer in his hand, he proves the theorem. And the teacher, by clicking the mouse, reveals each correct step of reasoning.
I stopped using this good algorithm. Because The projector in the classroom is on the desk. In this case, the projector beam shines into the child’s eyes, he closes his eyes and experiences discomfort. This is very harmful to the eyes! The optimal location for the projector is on the ceiling. Then the projector beam goes above our heads, and does not shine into our eyes. When inviting students to the board while the projector is on, choose a location away from the screen. Dear colleagues, take care of your eyes! Avoid direct eye contact with the projector beam.
On slides 14 -17 given game tasks. How to make such modules is described in the resource “Geometry. Using Presentations to Illustrate Definitions.” Using the time of recording the start of the animation using a trigger, you can make game modules. These little ones test tasks successfully offered at any stage of the lesson. The main thing is measure.
Author's technique. When studying many geometry topics, it is useful to assign “Paired Problems.” Again, the advantage of a presentation is that you can prepare the slide in advance. It is quite difficult to prepare such “pairs” on a chalk board for a lesson; it takes time.
The purpose of compiling “Paired Problems” is to systematize knowledge on the topic.
On slide 18 an example is given. Problems on the topic “Properties of a parallelogram” and “Characteristics of a parallelogram.” How to organize work?
Teacher. There are two tasks on the slide. In the first problem it is given: ABCD is a parallelogram, and in the second problem it is necessary to prove that ABCD is a parallelogram. In which problem will we need the properties of a parallelogram, and in which the characteristics of a parallelogram?Students. They give an answer.
We solve two problems orally. Pronouncing the wording of the applied properties.
Slide 19– homework problem No. 383.
Teacher. Here's your homework assignment. Let's figure out what you need to solve this problem: properties or characteristics of a parallelogram.
Students. Given a parallelogram ABCD, this means you can apply the properties of a parallelogram. To prove that APCQ is a parallelogram we will need parallelogram features.
My students immediately saw that it was possible to prove the equality of triangles ABP and CDQ, DQ and SVR using 1 sign of equality of triangles. Then, AP=CQ, PC=AQ, and if in a 4-gon the opposite sides are equal, then APCQ is a parallelogram.
But I had to show them another method, which is embedded in the slide animations. Then they realized that there was another way to prove that ABCQ is a parallelogram. Using the 3º sign, through the diagonals.
We discussed two ways to solve this problem at home.
Slide 20. Another example of pair problems. In the 7th grade, it is important to teach children to distinguish in which problems the signs of parallelism of lines will be required, and in which problems it is necessary to apply inverse theorems.
This slide provides a visual cue for paired tasks - the key difference between the tasks is highlighted in red on the slide. In the first problem, “AB II CD” is highlighted in color, and in the second problem, “a II b”. If you offer similar paired tasks in the next lesson, then you can no longer give visual cues with color.
Teacher. Key Difference between tasks are highlighted in color on the slide. The first task requires prove that the lines are parallel
. And in the second problem given two parallel lines
. Which problem will require signs of parallelism of lines? And what is the converse theorem - about the intersection of two parallel lines by a transversal?
We solve the first problem orally, with commentary. By the way, in the first problem you can justify the solution differently: on the basis of parallelism through one-sided angles.
We solve the second problem in a notebook. We begin to reason orally all together. If no one remembers that we solve such problems algebraically, denoting one part as “x,” then we display a visual hint for the accompanying hero: “Let x be 1 part.” Next, children will remember: then the angles are respectively equal to 5x and 4x, and the sum of one-sided angles at the intersection of two parallel straight thirds is equal to 180º. So we can create an equation.
Let (x)º – 1 part
I’ll create and solve an equation...
Comment. When writing solutions in a notebook, I often use abbreviations. For example, OU are one-sided angles, similarly, NLU, SU. Theorem on three perpendiculars of TTP, etc.
Slides 21 – 23. At the stage of preparation for a new theorem, you can create modules to organize repetition. An example from an 8th grade geometry course. To prove the theorem about the area of a trapezoid, I needed to remind the children about the property of areas. I decided to look at the problem from the textbook so that the children could then come up with a proof of the theorem themselves.
Slide 21. We repeated the property of areas. Using this property, you can calculate the areas of various figures by breaking them into parts.
Slide 22. Let's consider the problem from textbook No. 478. The slide shows how to construct a quadrilateral. It’s convenient to start building with diagonals! And then construct the sides of the quadrilateral. I never put visual cues on the screen; I listen to students' ideas first. One student suggested calculating the area for each of the four right triangles and then adding them together. Unfortunately, no other ideas were proposed. I invited the girl to the board, she solved the problem in her own way.
Again I invite the children to think. After all, you can consider other triangles and solve the problem easier. Now you've guessed it. The triangles were named KMB, VRK and MVR, MKR. The second option was discussed orally. Which way is more beautiful? The one we wrote down in our notebooks or the one the computer offers us? We made a choice. It is advantageous to break the figure into fewer parts. We started the drawing with diagonals, perhaps this prevented the children from thinking. But, nevertheless, we are prepared to understand the theorem about calculating the area of a trapezoid.
Slide 23. So, suggest a way to break the figure into parts for which we can find the area using the formulas known to us. They suggested diagonal BD or AC.
With commentary we look through the animations of additional constructions and proofs. Then right-click, select “black screen”. Complete the evidence in your notebook. One student is invited to the board.
Slides 24 – 29. Fragment of the lesson. Theorem on the ratio of the areas of triangles having each equal angle. Relevant knowledge: Corollary 2 about the ratio of the areas of triangles having equal heights. Slides 24, 25 updating knowledge. We repeated it and reinforced it with an example. On slide 25, we noticed that for triangle ABC the height lies in the inner region of the triangle, and for triangle FBR the height lies in the outer region. For example, you can ask children: how does the location of the height differ for each triangle?
The theorem has a very complex drawing. It is difficult for a teacher to draw on the board and at the same time provide individual assistance to children. It is more convenient to work on a theorem with a module prepared in advance. The teacher shows animations, working with a remote mouse, and at the same time works individually with students. We build a drawing and prove it together with the computer.
We stipulate that we will call the vertex A 1 A. Therefore, we write A 1 in parentheses. After each animation we ask the children a question. For example, the CH height appeared on the screen. For which triangles is this height common?... Answer. How to write the ratio of the area of triangle ABC to the area AB 1 C. Answer... We display the height CH 1 on the screen. For which triangles is this height common?... Answer. How to write the ratio of the area of triangle AB 1 C to the area AB 1 C 1. Answer... Multiply equalities... etc.
Slides 28, 29 to consolidate the proven theorem. Agree that it is difficult for a teacher to do all this work with chalk on a blackboard. This means that there is another important advantage of using modules: to make the teacher’s hard work easier.
Geometry
chapter 7
Prepared by Daria Kirillova, 9th grade student
Teacher Denisova T.A.
1.Definition of similar triangles
a) proportional segments
b) definition of similar triangles
c) Area ratio
a) The first sign of similarity
b) Second sign of similarity
c) The third sign of similarity
A) middle line triangle
b) Proportional segments V right triangle
c) Practical applications of triangle similarity
b) The value of sine, cosine and tangent for angles 30 0, 45 0 and 60 0
The relationship between segments AB and CD is called the ratio of their lengths, i.e. AB:CD
AB = 8 cm
CD = 11.5 cm
Segments AB and CD are proportional to segments A 1 IN 1 and C 1 D 1 , If:
AB= 4 cm
CD= 8 cm
WITH 1 D 1 = 6 cm
A 1 IN 1 =3 cm
Similar figures- these are figures same shape
If in triangles all angles are respectively equal, then the sides lying opposite equal angles are called similar
Let in triangles ABC and A 1 IN 1 WITH 1 the angles are respectively equal
Then AB and A 1 IN 1 ,VS and V 1 WITH 1 ,SA and C 1 A 1 -similar
Two triangles are called similar , if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other triangle
K- similarity coefficient
back
The sides of one triangle are 15 cm, 20 cm, and 30 cm. Find the sides of a triangle similar to this if the perimeter is 26 cm
The ratio of the areas of two similar triangles equal to the square of the similarity coefficient
Proof:
The similarity coefficient is equal to K
S and S 1 are the areas of triangles, then
According to the formula we have
The first sign of similarity of triangles
If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar
Prove:
Proof
1)By the theorem on the sum of the angles of a triangle
2) Let us prove that the sides of the triangles are proportional
Same with the corners
So the sides
proportional to similar sides
The second sign of similarity of triangles
If two sides of one triangle are proportional to two sides of another triangle and the angles between these sides are equal, then such triangles are similar
Prove:
Proof
The third sign of similarity of triangles
If three sides of one triangle are proportional to three sides of another, then such triangles are similar
Prove:
Proof
Middle line called a segment connecting the midpoints of its two sides
Theorem:
The midline of a triangle is parallel to one of its sides and equal to half of that side
Prove:
Proof
Theorem:
The medians of a triangle intersect at one point, which divides each median in a ratio of 2:1, counting from the vertex
Prove:
Proof
In triangle ABC, median AA 1 and BB 1 intersect at point O. Find the area of triangle ABC if the area of triangle ABO is equal to S
Theorem:
The altitude of a right triangle drawn from the vertex of a right angle divides the triangle into two similar right triangles, each of which is similar to the given triangle
Prove:
Proof
Theorem:
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
Prove:
Proof
Determining the height of an object:
Determine the height of a telegraph pole
From the similarity of triangles it follows:
Practical applications of triangle similarity
Determining the distance to an invalid point:
Sinus - ratio of the opposite leg to the hypotenuse in a right triangle
Cosine - ratio of adjacent leg to hypotenuse in a right triangle
Tangent- ratio of the opposite side to the adjacent side in a right triangle
0 , 45 0 , 60 0
The value of sine, cosine and tangent for angles of 30 0 , 45 0 , 60 0
Similarity
Slides: 9 Words: 230 Sounds: 0 Effects: 117Similarity of triangles. Solving problems using ready-made drawings, grade 8. Mathematics teacher of the first quarter category RMOU Obskaya secondary school Vodyanova E.A. Problem 1. Prove: ?ХZR ~ ?RYZ Z Y 40° X 40° R. Problem 2. ABCD - trapezoid Prove: ?BOC ~ ?DOA B C O A D. Problem 3. ABCD - trapezoid Prove: ?ABC ~ ?ACD B C A D Name the proportional segments. Problem 4. BD || AF Find: AC; AB C 2 cm B D 3 cm A F 12 cm. Problem 5. KM || FH Find: FH H 4 cm K 7 cm 5 cm F M L. Problem 6. Find: AB C 2 cm 1 cm D B 5 cm 10 cm A F. Problem 7. Find: BD B 2 cm F D 5.5 cm 2 cm A C. Problem 8. ABCD - parallelogram Find: BD B C 16 cm 12 cm 8 cm D A R F. - Similarity.ppt
Similarity of triangles
Slides: 12 Words: 480 Sounds: 0 Effects: 85Similar triangles. Proportional segments. Definition of similar triangles. The number k, equal to the ratio of similar sides of the triangles, is called the similarity coefficient. Ratio of areas of similar triangles. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. The bisector of a triangle divides the opposite side into segments proportional to the adjacent sides of the triangle. Signs of similarity of triangles. III sign of similarity of triangles If three sides of one triangle are proportional to three sides of another triangle, then such triangles are similar Given: ?ABC, ?A1B1C1, Prove: ?ABC ?A1B1C1. - Similarity of triangles.ppt
Similar triangles
Slides: 19 Words: 322 Sounds: 0 Effects: 72Geometry. Triangle. Let's remember. Similar figures. How are the figures similar? Form! Definition of similar triangles. Signs of similarity of triangles. The angles are respectively equal. C1. Similar sides. Proportional. Similarity coefficient “k”. Name the similarities. Equality of relations between similar parties. Which triangles are similar? Circles are always similar. Squares are always similar. Very interesting. Shadow from the pyramid. Shadow from a stick. A little more about triangles. Proportional segments in a triangle. Height of the triangle. The altitudes of the triangle intersect at one point O, called the orthocenter. - Similar triangles.ppt
Similarity of triangles grade 8
Slides: 6 Words: 164 Sounds: 0 Effects: 0Application of similarity in human life. 1 sign of triangle similarity. 2 sign of similarity of a triangle. 3 sign of similarity of a triangle. Problem No. 1. Sides a and d, b and c are similar. Problem No. 2. - Similarity of triangles, grade 8.ppt
“Similar triangles” 8th grade
Slides: 42 Words: 1528 Sounds: 2 Effects: 381Similar triangles. Table of contents. Proportional segments. Segments. IN Everyday life there are objects of the same shape. Definition of similar triangles. Task. Similar sides. Two triangles are called similar. Similarity of triangles. Ratio of areas of similar triangles. Theorem. Properties of similarity. Triangles have equal angles. Signs of similarity of triangles. First sign. Similar sides are proportional. Second sign. General side. Third sign. The middle line of the triangle. Middle line. Medians in a triangle. O – intersection of medians. - “Similar triangles” 8th grade.ppt
Geometry Similarity of triangles
Slides: 9 Words: 405 Sounds: 0 Effects: 0Educational theme of the project. Similar triangles. Signs of similarity of triangles. Creative theme of the project: Abstract. The project was prepared outside of school hours by 8th grade students. Implemented within the framework of 8th grade geometry on the topic “signs of similarity of triangles.” The project includes an information and research part. Analytical work with information systematizes knowledge about such figures. Didactic tasks will help control the degree of absorption educational material. Reflection? Questions: What does the concept of “similar triangles” mean? How to measure the height of large buildings, trees...? - Geometry Similarity of triangles.ppt
Geometry "Similar Triangles"
Slides: 36 Words: 1995 Sounds: 0 Effects: 191Similar triangles. Proportional segments. Property of the bisector of a triangle. Two triangles are called similar. Problem solving. Theorem on the ratio of areas of similar triangles. The first sign of similarity of triangles. The second sign of similarity of triangles. Sides of a triangle. The third sign of similarity of triangles. Mathematical dictation. Proportionality of the sides of an angle. Similarity of right triangles. Continuation of the sides. The middle line of the triangle. The two sides of the triangle are connected by a segment not parallel to the third. Proportional segments in a right triangle. - Geometry “Similar triangles”.ppt
Definition of similar triangles
Slides: 48 Words: 2059 Sounds: 0 Effects: 138Similar triangles. Uses in life. Definition of similar triangles. Table of contents. Proportional segments. Two triangles are called similar. Ratio of areas of similar triangles. The first sign of the similarity of triangles. The second sign of the similarity of triangles. The third sign of similarity of triangles. Triangle ABC. The sides of triangle ABC are proportional. The sides of triangle ABC are proportional to similar sides. Consider triangle ABC. ABC. Triangles ABC and ABC are equal on three sides. Practical applications of triangle similarity. - Definition of similar triangles.ppt
Signs of similarity
Slides: 24 Words: 618 Sounds: 0 Effects: 154Similar triangles. Signs of similarity of triangles. Definition of similar triangles. The first sign of similarity of triangles. Given. Prove: Proof: So, the sides of triangle ABC are proportional to similar sides of triangle A1B1C1. The second sign of similarity of triangles. 13. 16. The third sign of similarity of triangles. Proof of the theorem. Theorem: Given: ?ABC, ?A1B1C1 AB/A1B1=BC/B1C1=CA/C1A1. Taking into account the second criterion for the similarity of triangles, it is enough to prove that Similarity criteria.ppt
Signs of similarity of triangles
Slides: 8 Words: 224 Sounds: 0 Effects: 100Signs of similarity of triangles. 1. Sign of similarity of triangles at two angles. There are three signs of similarity: A in a1b1. 3. Sign of similarity of triangles on three sides. Similarity of right triangles. - Signs of similarity of triangles.ppt
Three signs of similarity of triangles
Slides: 75 Words: 2318 Sounds: 0 Effects: 117Similarity in geometry. Theme: "Similarity". Proportional segments. Two right triangles. Proportionality of segments. Similar figures. Figures of the same shape are called similar figures. Similar triangles. Two triangles are called similar if their angles are respectively equal. Similarity coefficient. Additional properties. Perimeter ratio. Common multiplier. Area ratio. Property of the bisector of a triangle. Bisector. The equation. Signs of similarity of triangles. The first sign of similarity of triangles. The angles of the triangles are respectively equal. Similar sides are proportional. - Three signs of similarity of triangles.ppt
Lesson Signs of similarity of triangles
Slides: 11 Words: 161 Sounds: 0 Effects: 91Geometry lesson “Signs of similarity of triangles.” Objective of the lesson: Generalization on the topic “Signs of similarity of triangles.” Lesson objectives: Similar figures. In similar figures the angles are equal. In such figures, the sides are proportional. Are the triangles similar? When. The first sign of similarity of triangles. If two sides of one triangle are proportional to two sides of another. Then such triangles are similar. The second sign of similarity of triangles. if the three sides of one triangle are proportional to the three sides of another, the third sign of similarity of triangles. - Lesson Signs of similarity of triangles.ppt
The first sign of similarity of triangles
Slides: 15 Words: 583 Sounds: 0 Effects: 163Blue light. Similarity of triangles. The first sign of similarity. Let's depict: What is the difference between the figures in each presented pair? Definition. The proportionality coefficient is called the similarity coefficient. What do you mean what? Is ABC similar to a triangle? A1B1C1? The angles are equal. The sides are proportional. Similarity, resemblance. Indicate the proportional sides. The sides of the triangle are 5 cm, 8 cm and 10 cm. In similar triangles ABC and A1B1C1 AB = 8 cm, BC = 10 cm, A1B1 = 5.6 cm, A1C1 = 10.5 cm. Physical education: Do it all at once Repeat four times . 2. Set aside: segment AB"= A1B1 (point B" є AB) straight line B"C" || Sun. - The first sign of similarity of triangles.ppt
Ratio of areas of similar triangles
Slides: 6 Words: 250 Sounds: 0 Effects: 35Similar triangles. Content. Similar figures. In everyday life, there are objects of the same shape, but of different sizes. In geometry, figures of the same shape are called similar. The number k, equal to the ratio of similar sides of the triangles, is called the similarity coefficient. The ratio of the perimeters of similar triangles. The ratio of the perimeters of two similar triangles is equal to the similarity coefficient. Ratio of areas of similar triangles. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. - Ratio of areas of similar triangles.ppt
Application of similarity
Slides: 11 Words: 457 Sounds: 0 Effects: 9Application of similarity to problem solving. 8th grade. Conversation. Option 1 Determine similar triangles. Formulate the third criterion for the similarity of triangles. State the bisector property of a triangle. Option 2 Determination of the midline of the triangle. Formulate the first sign of similarity of triangles. State the property of the intersection point of the medians of a triangle. Oral work. What fraction of the area of triangle ABC is the area of trapezoid AMNC? Problem solving. Calculate the medians of a triangle with sides 25 cm, 25 cm and 14 cm. O is the point of intersection of the diagonals of the parallelogram ABCD, E and F are the midpoints of the sides AB and BC, OE = 4 cm, OF = 5 cm. - Application of similarity.ppt
Application of triangle similarity
Slides: 8 Words: 127 Sounds: 0 Effects: 29Practical application of triangle similarity. Lesson plan. Application of similarity of triangles in proving theorems. Construction tasks. Measurement work on the ground. Triangle midline theorem. Property of medians of a triangle. Proportional segments in a right triangle. Division of a segment in a given ratio. Construction of triangles. Divide the segment in a ratio of 2/3. Determining the height of an object. Determining the distance to an inaccessible point. Determining the height of an object using a mirror. - Application of similarity of triangles.ppt
Application of the similarity of triangles in life
Slides: 31 Words: 1146 Sounds: 0 Effects: 12Practical application of triangle similarity. Similarity in life. A little bit of history. The rod is approximately the height of a man. Determining the height of an object. Determining the height of the pyramid. Historical reference. Tired stranger. Thales. Thales' method. Shadow from a stick. Determining the height of an object using a pole. Mysterious Island. Finding the fourth unknown term of the proportion. Determining the height of an object from a puddle. Determining the height of an object using a mirror. Advantages. Determining the distance to an inaccessible point. Finding the width of the lake. Distance to tree. Pin measuring device. - Application of the similarity of triangles in life.ppt
Practical application of triangle similarity
Slides: 16 Words: 530 Sounds: 0 Effects: 0practical application of triangle similarity. Fairy tale. Shrek's birthday. Shrek came home. Geometry lessons. Similarity of triangles. Everything was decided correctly. The distance from one shore to the other. You can use the similarity of triangles. Solution. Rope of the required length. Idea. Bracelet. - Practical application of triangle similarity.pptx
Practical applications of triangle similarity
Slides: 10 Words: 454 Sounds: 0 Effects: 0Topic: Practical applications of triangle similarity. Creative name: Determining the height of an object. How can you measure the height of an object using simple devices? What methods are there to determine the height of an object? What instruments or devices are needed to measure the height of an object? What are the similarities and differences in determining the height of an object? Study topic question: Application of similarity of triangles. Academic subjects: geometry, literature, physics. Participants: 8th grade students. Presentation-abstract, booklet, newsletter on methods for determining the height of an object. - Practical applications of similarity of triangles.ppt
Problems like
Slides: 21 Words: 436 Sounds: 0 Effects: 1Solving geometry problems using ready-made drawings. Task topics. The first sign of similarity of triangles. The second and third signs of similarity of triangles. Similar triangles. Example No. 2. Example No. 1. Example No. 4. Example No. 3. Example No. 6. Example No. 7. Example No. 5. - Similar problems.ppt
Problems similar to triangles
Slides: 38 Words: 1448 Sounds: 0 Effects: 48Similarity of triangles. The first sign of similarity. What triangles are called similar. Formulate the first sign of similarity of triangles. The triangles shown in the figure. Draw a triangle. Triangle. Sides of a triangle. Right triangles. The two triangles are similar. Sides of triangles. Perimeter. List all similar triangles. Side. Square. Vertex. Is it possible to intersect a triangle with a straight line? Chords of a circle. Find similar triangles. Acute triangle. Product of segments. Radius of a circle. Circle. Two straight. - Problems similar to triangles.ppt
Similarity of triangles problem solving
Slides: 6 Words: 331 Sounds: 0 Effects: 0Similar triangles. The concept of similarity is one of the most important in the planimetry course. The study of the topic begins with the formation of the concepts of the relationship of segments and the similarity of triangles. Solving construction problems using the similarity method is discussed with students interested in mathematics. This topic is intended for 8th grade students. 19 hours are allocated for studying the material. Lesson topic: The first sign of similarity of triangles. Checking homework. Solving problems to prepare students to perceive new material. Learning new material. Formulation of 1 criterion for the similarity of triangles. Proof of the theorem. - Similarity of triangles problem solving.ppt
Triangle similarity problems
Slides: 22 Words: 326 Sounds: 0 Effects: 48Similarity of triangles. Lesson motto. Individual card. Name similar triangles. Solving practical problems. Determining the height of the pyramid. Thales' method. Shadow from a stick. Measuring the height of large objects. Determining the height of an object. Determining the height of an object using a mirror. Determining the height of an object from a puddle. Solving problems using ready-made drawings. Gymnastics for the eyes. Independent work. -
Slide 2
GAME STRUCTURE 1 race 2 race 3 race 4 race 5 race Hurray!!! “Further..., further..., further...” “You are for me, I am for you” “To the past in a time machine” “Troubles from the pot” “You and only you” Summing up
Slide 3
“Further..., further..., further...” First command Second command How to continue the statement so that it becomes true? “If two angles of one triangle...” 1 Continue the phrase so that the statement becomes true. “The leg of a right triangle is ...” KNOW!!!
Slide 4
First team Second team 2 Think!!! Given: ABCD-parallelogram. Find: similar triangles to prove their similarity. Next... Given: DE║AC. Find:X. A B F C D K A B C D E X 3 6 12 Fig. 1 Fig. 2
Slide 5
First team Second team 3 Apply!!! Next... Given: ∆ABC ∆MNK. Find: x, y. S Given: DC ┴ AB,AE ┴ BC. Is it true that ∆BAE ∆BCD ? S A A B B C C M N K 8 4 x y 4 3 D E Fig. 3 Fig. 4
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First team Second team 4 Figure it out!!! Next... Let BC║AD. Write down the proportional segments. Given: AB·BK = CB·BP. Find equal angles, if any. Rice. 5 Fig. 6 A B C D A B C K P
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First team Second team 5 Get tense!!! Next... Given: MNKF-rectangle. How many similar triangles were formed? Are the drawn triangles similar? A B C M N K F 43° 73° 43° 64° Fig. 7 Fig. 8
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"You - for me, I - for you"! ! ! ? ? ?
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“To the past in a time machine” Ancient Greece Miletus Money Men's suit Ancient Egypt Measured the height of the pyramid without climbing onto it. Who is he??? Lived 640-548 BC Numbered among the SEVEN SAGES OF THE LIGHT. He owns the aphorism: “Know yourself.” Started a game of "PROVED". Entered calendar: 1 year = 365 days
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Sunlight B C dimension shadow K E D Θαλῆςὁ Μιλήσιος Fig. 9 A "How Thales measured the height of the pyramid"
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Angle of view pole rock Fig. 10 ? 10 15 500 “Troubles from the pot” Problem 1. The method of Jules Verne (travel writer) 1828-1905
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Problem 2. Lumberjacks' method for determining the height of trees that cannot be accessed Instruments for constructing a visual angle 2X 2X X Two boards 2X 2X 2X X Visual angle Visual angle Notepad and pencil 2X 2X X 2X M F h A K B D E C H N Fig. eleven
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"You and only you" Fig. 12 A B C D E M O F Given:BD║AE. Name pairs of similar triangles. Formulate a well-known theorem, the proof of which uses this geometric construction. Given: lengths of segments a and b. Using a compass and ruler, construct a segment X - the geometric mean of the lengths of segments a and b. Are any two isosceles triangles similar? 3 1 2
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“You and only you” The lengths of the segments a, b and c are given. The segments b and c lie on the same straight line. How can we construct X = a b/c using this geometric construction, where X is called the fourth proportional? c b a Fig. 13 4 5 Is it possible to intersect two sides of a triangle with a straight line, not parallel to the third side, so that it cuts off a triangle similar to the original one? ║ ║
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THANK YOU TO ALL FURTHER CREATIVE SUCCESS!
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Internet sources 2. Ancient Greece 1. Sound (birds singing, sound of the sea surf) http://wav.wizardsound.ru/main/sounds/animals/ http://wav.wizardsound.ru/main/sounds/nature / http://afield.org.ua/mod3/mod40_2.htmlhttp://www.vrata11.ru/gallery/turkey5.htm http://ru.wikipedia.org/w/index.php?title=%D0 %A4% D0%B0%D0%BB%D0%B5%D1%81&redirect=no http://pavlov-museum.narod.ru/antiq/index.html http://history.rin.ru/text/tree /124.html http://history.rin.ru/cgi-bin/history.pl?num=3645
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http://www.3dnews.ru/editorial/it_apocalypse/ http://www.detfond.org/cover.php?izdanie=classic&id=36 http://my-shop.ru/shop/books/154411.html http://innatour.ur.ru/Izrail/o_strane/eylat_kruiz.htm 3. Ancient Egypt 4. Jules Verne http://www.morev.de/wonders/classic/piramides.htmlhttp://afield.org.ua /ist/neit.html http://helen.org.ua/photo/gallery/thumbnails.php?album=10 http://www.tmn.fio.ru/works/101x/311/102.htm
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Geometry
chapter 7
Prepared by Namazgulova Gulnaz, 8b grade student of the State Budgetary Educational Institution RPLI in Kumertau
Teacher: Bayanova G.A.
The relationship between segments AB and CD is called the ratio of their lengths, i.e. AB:CD
AB = 8 cm
CD = 11.5 cm
Segments AB and CD are proportional to segments A 1 IN 1 and C 1 D 1 , If:
CD= 8 cm
AB= 4cm
WITH 1 D 1 = 6 cm
A1B1=3 cm
Two triangles are called similar , if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other triangle
K- similarity coefficient
The ratio of the areas of two similar triangles equal to the square of the similarity coefficient
Proof:
The similarity coefficient is equal to K
S and S 1 are the areas of triangles, then
According to the formula we have
The first sign of similarity of triangles
If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar
Prove:
Proof
1)By the theorem on the sum of the angles of a triangle
2) Let us prove that the sides of the triangles are proportional
Same with the corners
So the sides
proportional to similar sides
The second sign of similarity of triangles
If two sides of one triangle are proportional to two sides of another triangle and the angles between these sides are equal, then such triangles are similar
Prove:
Proof
The third sign of similarity of triangles
If three sides of one triangle are proportional to three sides of another, then such triangles are similar
Prove:
Proof
Middle line called a segment connecting the midpoints of its two sides
Theorem:
The midline of a triangle is parallel to one of its sides and equal to half of that side
Prove:
Proof
Theorem:
The medians of a triangle intersect at one point, which divides each median in a ratio of 2:1, counting from the vertex
Prove:
Proof
Theorem:
The altitude of a right triangle drawn from the vertex of a right angle divides the triangle into two similar right triangles, each of which is similar to the given triangle
Prove:
Proof
Theorem:
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
Prove:
Proof
Sinus - ratio of the opposite leg to the hypotenuse in a right triangle
Cosine - ratio of adjacent leg to hypotenuse in a right triangle
Tangent- ratio of the opposite side to the adjacent side in a right triangle
0 , 45 0 , 60 0
The value of sine, cosine and tangent for angles of 30 0 , 45 0 , 60 0
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