Theorem on the middle line of the trapezium. Middle line trapezium middle line trapezium presentation
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The theme "The middle line of the trapezium" refers to one of the important topics of the geometry. This figure is often found in various tasks, like its middle line. Tasks containing the data of this topic are often found in final control and certification work. Knowledge on this topic may also come in handy in training in medium and higher establishments.
Although the topic declared a trapezoid figure, but the consideration of this topic can pass in the period of studying the topic "Vectors" and "Application of vectors when solving tasks." This can be understood by looking at the presentation slide.
The author here defines the average line as a segment that connects the middle of the side. Moreover, it also noted that the middle line of the trapezium is parallel to its grounds, and is also equal to their half asch. It is in the course of the proof of this statement and knowledge of the knowledge associated with vectors. Applying the rules for the addition of vectors according to the drawing, which is shown as an illustration of the condition, the equality is obtained. These equalities have the same left part, and it is the middle line of the trapezoid in the form of a vector. Folding these equality, a large expression is obtained in the right part of equality.
slides 1-2 (the topic of the presentation "middle line of the trapezoid", the definition of the middle line of the trapezium)
If you consider carefully, then in two cases it turns out the addition of opposite vectors giving zero. Then it remains that the double vector containing the average line of the trapezoid is equal to the sum of the vectors containing the bases. Sharing this equality by 2, it turns out that the vector containing the average line is equal to half the sum of vectors containing bases. Now there is a comparison of vectors. It turns out that all these vectors are equally directed. This means that the signs of the vectors can be safely omitted. And then it turns out that the middle line of the trapezium is equal to the middle of the base.
The presentation contains a single slide that carries a large amount of information. Here is given the definition of the middle line of the trapezium, as well as its main property. In the course of geometry, this property is the theorem. So here is proved the theorem using the knowledge of the concept of vectors and actions above them.
The teacher can add this presentation to its examples and tasks, but everything that is required for the average level of knowledge on this subject is published here. Moreover, so the author left the opportunity to make a teacher to fantasize, refine what he wanted himself in order to create an appropriate atmosphere in the lesson. Do not forget about the mood itself on the lesson. Then, with the help of this presentation, it is precisely possible to achieve the desired result.
Definition: The middle line of the triangle is called a segment connecting the middle of his two sides. AK \u003d COP \u003d CE KE - the average line of ABC Definition: The middle line of the trapezium is called a segment connecting the middle of the side of it. And the sun KN \u003d CE \u003d NV KE \u003d CE is not - the middle line Avsk and in C to e how many medium lines in the triangle? How many medium lines in the trapezium?
The middle line of the triangle theorem. The middle line of the triangle is parallel to one of its sides and is equal to half of this side. And with in m to given: ABC, MK - the middle line Proof: T. K. under the condition MK - the middle line, then AM \u003d MV \u003d ½ AB, SK \u003d KV \u003d ½ Sun, it means that VM AV VK Sun 1 2 - General for ABC and MVK, it means that ABC and MVK are similar to the second sign of the likeness, therefore, IMK \u003d A, it means that MK AU. Prove: MK AF, MK \u003d ½ AC MK AC 1 2 from the similarity of the triangles also follows that, that is, MK \u003d ½ speavers.
Share the problem F R n? A B.
Proof: Conduct a 1 in 1 A B with A1A1 B1V1 O C1C1 under the condition AA 1, BB 1 - medians means, VA 1 \u003d CA 1, AV 1 \u003d SV 1, i.e. and 1 in 1 - the middle line. So, 1 in 1 AB, therefore 1 \u003d 2, 3 \u003d 4. Therefore, the triangles of AOs and a 1 s 1 are like two corners. It means that their parties are proportional to: A1OA1O1O1O1O1O1B1A1B1A1B1A 1 in the property of the middle line of the triangle AV \u003d 2 A 1 in 1, i.e., A1O1O1O1B1 1 1 1 1 1 1 is similar, with C1O1O 2 1, we get: C1O1O1O 2 1
The average line of the theorem trapezium. The middle line of the trapezium is parallel to the grounds and is equal to half a half. And in C to M R Dano: Avsk - a trapezium of MR - the middle line to prove: MR AK, MR Sun MR \u003d Proof: ON PRESSIBLE TO THE POINT MR Direct Me AK, we will prove that I will pass through R. T.K. Avsk - trapezium , then the aircraft AK, and, it means that the MP is the middle line, then AM \u003d MV, the cr \u003d cf is therefore, MR lies on me, it means that MR AK, MR Sun. Cut the VC. According to the Falez Theorem O - Mid-VC, it means that the MO is the average line of AVC, OR - the middle line SPA MR \u003d MO + OP \u003d ½ AK + ½ Sun \u003d ½ (AK + Sun) \u003d on the Fals Theorem ME will cross the SC in the middle of the SC, i.e. at the point R.
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"Falez Theorem" - Fales is widely known as a geometer. Astronomy. Miletsky materialist. We carry out a direct EF direct EF point, parallel to the direct A1A3. Of the equality of triangles, the equality of the parties is B1B2 \u003d B2B3. Falez Theorem. It is believed that Falez first studied the movement of the sun on the heavenly sphere. Triangles B2B1F and B2B1e are equal on the second basis of the equality of triangles.
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Signatures for slides:
Medium line (grade 8)
The middle line of the triangle
The middle line of the triangle. Definition: The segment connecting the middle of the two sides of the triangle is called the middle line of the triangle.
The theorem The middle line of the triangle is parallel to one of its sides and is equal to half of this side. Those.: KM ║ AS km \u003d ½ AC A B C K M
Solve the problem orally: a b c k m 7 cm is given: M to - environments. Line Find: AS?
Work in pairs:
We will solve the task: given: Mn - media. Line Find: P Δ ABS M n A B C 3 4 3, 5
Work in pairs:
Medium line trapezium
Recall: The trapezium is a quadrangle, in which two sides are parallel, and the other two parties are not parallel to A D B C BC || AD - Bases AB łł CD - Side
The middle line of the trapezium. Definition: The middle line of the trapezium is called a segment connecting the middle of its side sides. A D B C M N Mn - ABCD Middle Line
The average trapezium line the average line of the trapezium is parallel to its bases and is equal to half a semit. Those.: M n ║ves d m n \u003d ½ (Sun + A D) M n A D B C
Solve orally: M n a d b c 6.3 cm 18,7 cm?
Solve orally in pairs: AB \u003d 16 cm; Cd \u003d 1 8 cm; M n \u003d 15 cm Find: p abcd \u003d? M n a d b c
Independent work Task: The average trapezium line is 5 cm. Find the base of the trapezoid, if it is known that the lower base is more than 1.5 times. Solution: A D B C 5 cm Suppose Bc \u003d x cm then ad \u003d 1.5x cm BC + ad \u003d 10 cm x + 1.5x \u003d 10 x \u003d 4 means: bc \u003d 4 cm ad \u003d 6 cm
THANK YOU FOR THE LESSON!!!
The presentation was developed by Mathematics Teacher GBOU SOSH No. 467. St. Petersburg, Kolpinsky district Lugvina Natalia Anatolyevna
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