Download presentation on the topic of similarity of triangles. Similarity of right triangles. Practical applications of triangle similarity
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) Midline of the triangle
b) Proportional segments in a 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 of the 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
Let us depict: a) two unequal circles; b) two unequal squares; c) two unequal isosceles right triangles; d) two unequal equilateral triangles. a) two unequal circles; b) two unequal squares; c) two unequal isosceles right triangles; d) two unequal equilateral triangles. How are the figures in each pair presented different? What do they have in common? Why are they not equal?
In similar triangles ABC and A 1 B 1 C 1 AB = 8 cm, BC = 10 cm, A 1 B 1 = 5.6 cm, A 1 C 1 = 10.5 cm. Find AC and B 1 C 1. A B C A1A1 B1B1 C1C,6 10.5 similar,6 10.5 x y Answer: AC = 14 m, B 1 C 1 = 7 m.
Physical education lesson: The lesson has been dragging on for a long time. You have decided a lot. The bell will not help here, Since your eyes are tired. We do everything at once. We repeat four times. – Follow the similarity sign with your eyes. - Close your eyes. – Relax your forehead muscles. – Slowly move your eyeballs to the far left position. – Feel the tension in your eye muscles. – Fix the position – Now slowly, with tension, move your eyes to the right. – Repeat four times. - Open your eyes. – Follow the similarity sign with your eyes.
The first sign of similarity Theorem. (The first sign of similarity.) If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar. A B C C1C1 B1B1 A1A1 C"C" B"
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
“Similarity problems” - Similar triangles. Find x, y, z. Example No. 4. Solving geometry problems using ready-made drawings. Problem condition: Given: ?ABC ~ ?A1B1C1. Task topics. Example No. 2. Author: Skurlatova G.N. Municipal educational institution "Secondary school No. 62". The first sign of similarity of triangles. End the presentation. Example No. 1. The second and third signs of similarity of triangles.
“Lesson Signs of similarity of triangles” - In similar figures, the sides are proportional. A. A1. Geometry lesson “Signs of similarity of triangles.” IN 1. Objective of the lesson: Generalization on the topic “Signs of similarity of triangles.” When. B. In similar figures, the angles are equal. Similar figures. Lesson Objectives: Are triangles similar?
“Practical applications of triangle similarity” - What methods exist for determining the height of an object? Study topic question: Application of similarity of triangles. Presentation-abstract, booklet, newsletter on methods for determining the height of an object. How can you measure the height of an object using simple devices? Academic subjects: geometry, literature, physics.
“Signs of similarity” - A. Similar triangles. C. ABC and A1 B1C1 are triangles<А=А1; <В=<В1. C1. B. Дано. 4. Признаки подобия треугольников. 3. 1. 2. “Similarity of triangles, grade 8” - 1 sign of the similarity of a triangle. Prepared by 8th grade student Dmitry Mikhalchenko. 3 sign of similarity of a triangle. Problem No. 1. 2 sign of similarity of a triangle. Sides a and d, b and c are similar. Application of similarity in human life. “Application of similarity of triangles” - Proportional segments in a right triangle. Division of a segment in a given ratio. Divide the segment in a ratio of 2/3. Practical application of triangle similarity. B. Application of similarity of triangles in proving theorems. Measurement work on the ground. Triangle midline theorem. 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. The free time that has appeared 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. 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. 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 game tasks are given. 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 small test tasks can be 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? 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. The key differences between the 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 equal angles. 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.
- 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 conducts 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.
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.
Students. They give an answer.
We solve two problems orally. Pronouncing the wording of the applied properties.