Lesson 34 THEOREM. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. where k is the similarity coefficient. The ratio of the perimeters of two similar triangles is equal to the similarity coefficient. V. A. S. R. M. K. Problem solving: No. 545, 549. Homework: p. 56-58, No. 544, 548.
slide 6 from the presentation "Geometry "Similar Triangles"". The size of the archive with the presentation is 232 KB.Geometry Grade 8
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The purpose of the lesson: give a definition of similar triangles, prove the theorem on the ratio of similar triangles.
Lesson objectives:
- Educational: students should know the definition of similar triangles, the theorem on the ratio of similar triangles, be able to apply them in solving problems, implement interdisciplinary connections with algebra and physics.
- Educational: to cultivate diligence, attentiveness, diligence, to cultivate a culture of behavior of students.
- Developing: development of students' attention, development of the ability to reason, think logically, draw conclusions, develop students' competent mathematical speech and thinking, develop skills of introspection and independence.
- Health saving: observance of sanitary and hygienic standards, change of activities in the lesson.
Equipment: computer, projector, didactic material: independent and control work in algebra and geometry for grade 8 A.P. Ershova, etc.
Lesson type: learning new material.
During the classes
I. Organizational moment(greeting, checking readiness for the lesson).
II. The topic of the lesson.
Teacher: In everyday life there are objects of the same shape, but different sizes.
Example: soccer and tennis balls.
In geometry, figures of the same shape are called similar: any two circles, any two squares.
Let us introduce the concept of similar triangles.
Definition: 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.
Number k, equal to the ratio of similar sides of similar triangles is called the coefficient of similarity. ∆ABC ~ A 1 B 1 C 1
1. Orally: Are triangles similar? Why? (prepared drawing on the screen).
a) Triangle ABC and triangle A 1 B 1 C 1 if AB = 7, BC = 5, AC = 4, ∠A = 46˚, ∠C = 84˚, ∠A 1 = 46˚, ∠B 1 = 50 ˚, A 1 B 1 \u003d 10.5, B 1 C 1 \u003d 7.5, A 1 C 1 \u003d 6.
b) In one isosceles triangle, the angle at the apex is 24˚, and in the other isosceles triangle, the angle at the base is 78˚.
Guys! Recall the theorem on the ratio of the areas of triangles having an equal angle.
Theorem: If the angle of one triangle is equal to the angle of another triangle, then the areas of these triangles are related as the products of the sides containing equal angles.
2. Written work according to prepared drawings.
On-screen drawing:
a) Given: BN: NC = 1:2,
BM=7cm, AM=3cm,
S MBN \u003d 7 cm 2.
Find: S ABC
(Answer: 30 cm2.)
b) Given: AE = 2 cm,
S AEK \u003d 8 cm 2.
Find: S ABC
(Answer: 56 cm2.)
3. Let us prove the theorem on the ratio of the areas of similar triangles ( the student proves the theorem on the blackboard, the whole class helps).
Theorem: The ratio of two similar triangles is equal to the square of the similarity coefficient.
4. Actualization of knowledge.
Problem solving:
1. The areas of two similar triangles are 75 cm 2 and 300 cm 2. One of the sides of the second triangle is 9cm. Find the side of the first triangle that is similar to it. ( Answer: 4.5 cm.)
2. Similar sides of similar triangles are 6 cm and 4 cm, and the sum of their areas is 78 cm 2. Find the areas of these triangles. ( Answer: 54 cm2 and 24 cm2.)
If there is time independent work educational character.
Option 1
Similar triangles have similar sides equal to 7 cm and 35 cm.
The area of the first triangle is 27 cm2.
Find the area of the second triangle. ( Answer: 675 cm2.)
Option 2
The areas of similar triangles are 17 cm 2 and 68 cm 2. The side of the first triangle is 8 cm. Find the similar side of the second triangle. ( Answer: 4 cm)
5. Homework: geometry textbook 7-9 L.S. Atanasyan and others, pp. 57, 58, no. 545, 547.
6. Summing up the lesson.
Type of lesson: lesson of acquaintance with new material.The purpose of the lesson: To prove the property of the areas of similar triangles and show its practical significance in solving problems.
Lesson objectives:
teaching - to prove the property of the areas of similar triangles and show its practical significance in solving problems;
developing - to develop the ability to analyze and select arguments when solving a problem, the method of solving which is unknown;
educational - to cultivate interest in the subject through the content of the educational process and the creation of a situation of success, to cultivate the ability to work in a group.
The student has the following knowledge:
The unit of activity content that students need to learn:
During the classes.
1. Organizational moment.
2. Actualization of knowledge.
3. Dealing with a problem situation.
4. Summing up the lesson and recording homework, reflection.
Teaching methods: verbal, visual, problem-search.
Forms of training: frontal work, work in mini-groups, individual and independent work.
Technologies: task-oriented, information technologies, competency-based approach.
Equipment:
a computer, a projector for demonstrating a presentation, an interactive whiteboard, a document camera;
computer presentation in Microsoft PowerPoint;
reference summary;
During the classes
1. Organizational moment.
Today at the lesson we will work not in notebooks, but in supporting notes, which you will fill out for the duration of the entire lesson. Sign it. The assessment for the lesson will consist of two components: for the reference notes and for active work in the lesson.
2. Actualization of students' knowledge. Preparation for active educational and cognitive activity at the main stage of the lesson.
We continue to study the topic "similarity of triangles". So let's remember what we learned in the last lesson.
Theoretical workout. Test. In your reference notes, the first task has a test character. Answer the questions by choosing one of the suggested answers, where necessary, enter your answer.
Teacher: What is the ratio of two segments?
Answer: The ratio of two segments of two segments is the ratio of their lengths.
Teacher: In what case are the segmentsAB and CDproportional to segmentsA 1 B 1 and C 1 D 1
Answer: cuts AB and CDproportional to segmentsA 1 B 1 and C 1 D 1 if
your options. Good. Don't forget to correct whoever is wrong.
Teacher: What is the definition of similar triangles? Refer to your reference abstract. You have three answers to this question. Choose the right one. Circle it.
So, please, which option did you choose _______
Answer: Two triangles are called similar if their angles are respectively equal and the sides of one triangle are proportional to the sides of the other triangle.
Well done! Correct whoever is wrong.
Teacher: What is the ratio of the areas of two triangles that have the same angle?
Answer: If the angle of one triangle is equal to the angle of another triangle, then the areas of these triangles are divided as the products of the sides containing equal angles.
Solution of problems according to ready-made drawings.Further, our warm-up will take place in the course of solving problems according to ready-made drawings. You also see these tasks in your reference notes.
Reflection. Let's clarify what knowledge and skills allowed us to solve these problems. What solution methods did we use (fixing the answers on the board).
Possible answers:
Definition of similar triangles;
Application of the definition of similar triangles in solving problems;
Theorem on the ratio of the areas of triangles having an equal angle;
And now I propose a method for solving several problems that resonates with the topic of the lesson, but they are more related to geography.
situation of success.
The first task is in front of you. We are working on this issue on our own. The first one who succeeds will show his solution at the blackboard, and someone will demonstrate his solution through a document camera, so we write beautifully and accurately.
Answer: the sides of the Bermuda Triangle are 2000 km, 1840 km, 2220 km. The length of the border is 6060 km.
Reflection.
Possible answer: Similar triangles have similar sides that are proportional.
situation of success.
We figured out the dimensions of the Bermuda Triangle. Well, now let's find out the measurements of the flower bed. Flipping the base notes. Second task. We solve this problem by working in pairs. We check in a similar way, but only the result will be the first pair that has completed the task.
Answer: the sides of a triangular flower bed are 10m and 11m 20 cm.
So, let's check in. Does everyone agree? Who decides in a different way?
Reflection.
What course of action did you use to solve this problem? Record in your master note.
Possible answer:
similar triangles have corresponding angles equal;
The areas of triangles with equal angles are divided as the products of the sides containing equal angles.
Failure situation.
5. Learning new material.
When solving the third task, students are faced with a problem. They fail to solve the problem, because in their opinion the condition of the problem is not complete enough or they receive an unreasonable answer.
Students have not encountered this type of problem before, so there was a failure in solving the problem.
Reflection.
What method did you try to solve?
Why didn't you solve the last equation?
Pupils: We cannot find the area of a triangle if only the area of a similar triangle and the coefficient of similarity are known.
Thus, the purpose of our lesson find the area of a triangle if only the area of a similar triangle and the coefficient of similarity are known.
Let's reformulate the problem in geometric language. Let's solve it, and then return to this problem.
Conclusion: The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
Well, now let's go back to problem number 3 and solve it, based on a proven fact.
7. Summary of the lesson
What did you learn to do today?
Solve problems in which the coefficient of similarity and the area of one of the similar triangles are known.
What geometric property helped us in this?
The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
Homework.
P. 58 p.139 No. 546, 548
Creative task.
Find what is the ratio of the perimeters of two similar triangles (№547)
Goodbye.
1.3. The ratio of the areas of similar triangles. Theorem. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. Proof. Let triangles ABC and A1B1C1 be similar and the similarity coefficient be equal to k. Let S and S1 denote the areas of these triangles. Since A= A1, then.
slide 11 from the presentation ""Similar Triangles" Grade 8". The size of the archive with the presentation is 1756 KB.Geometry Grade 8
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