“Construction of regular polygons” - ?=60?. ·180?. Geometry. ?=. n. n - 2. The work was carried out by the mathematics teacher of the municipal educational institution “Gymnasium No. 11” Lisitsyna E.F.
"Thales' Theorem" - Thales's Theorem. A geometric theorem is named after Thales. Astronomy. Let us draw a line EF through point B2, parallel to line A1A3. It is believed that Thales was the first to study the movement of the Sun across the celestial sphere. Presentation on geometry by Polina Sorogina, class 9 “A” student. Milesian materialist. Geometry. According to the property of a parallelogram, A1A2 = FB2, A2A3 = B2E. Thales is widely known as a geometer. And since A1A2 = A2A3, then FB2 = B2E.
“Decomposition of a vector into two non-collinear ones” - Let p be collinear with b. Proof: Decomposition of a vector into two non-collinear vectors. Proof: Let a and b be non-collinear vectors. Lemma: If vectors a and b are collinear and a? 0, then there is a number k such that b = ka. Let us prove that any vector p can be decomposed into vectors a and b. Geometry 9th grade. Then p = yb, where y is a certain number.
“Regular polygons 9th grade” - Geometry lesson in 9th grade. Lukovnikova N.M., mathematics teacher. Constructing a regular pentagon 1 way. Municipal educational institution gymnasium No. 56, Tomsk-2007. Regular polygons.
“Symmetry of figures” - Line a is called the axis of symmetry of the figure. D. One figure is obtained from another by transformation. Table of contents. A transformation that is the opposite of a movement is also a movement. A1. Completed by: Pantyukov E. A. There are many different types of symmetry. M1. Transforming shapes.
“Symmetry relative to a straight line” - A figure can have one or more axes of symmetry. Symmetry in nature. Savchenko Misha, 9B grade. Corner. Who is shown in the original photograph? L.S. Atanasyan "Geometry 7-9". Isosceles trapezoid. Construct a segment A1B1 symmetrical to a segment AB relative to a straight line. How many axes of symmetry does each figure have? Rectangle.
The topic “The midline of a trapezoid” is one of the important topics in the geometry course. This figure is quite common in various problems, as is its middle line. Assignments containing data on this topic are often found in final tests and certification papers. Knowledge on this topic can also be useful when studying in secondary and higher institutions.
Although the topic includes a trapezoidal figure, consideration of this topic can take place during the period of studying the topic “Vectors” and “Application of vectors in solving problems.” This can be understood by looking at the presentation slide.
The author here defines the midline as a segment that connects the midpoints of the sides. Moreover, it is also noted here that the midline of the trapezoid is parallel to its bases and is also equal to their half-sum. It is precisely in the course of proving this statement that knowledge related to vectors will come in handy. Applying the rules for adding vectors according to the drawing, which is shown as an illustration of the condition, equalities are obtained. These equalities have the same left side, and it is the midline of the trapezoid as a vector. Adding these equalities, we get a large expression on the right side of the equality.
slides 1-2 (Presentation topic "Midline of the trapezoid", definition of the midline of the trapezoid)
If you look closely, in two cases you get the addition of opposite vectors, resulting in zero. Then it remains that the double vector containing the midline of the trapezoid is equal to the sum of the vectors containing the bases. Dividing this equality by 2, it turns out that the vector containing the middle line is equal to half the sum of the vectors containing the bases. Now comes the comparison of vectors. It turns out that all these vectors are equally directed. This means that the vector signs can be safely omitted. And then it turns out that the middle line of the trapezoid itself is equal to half the sum of the bases.
The presentation contains a single slide that contains a large amount of information. Here the definition of the midline of a trapezoid is given, and its main property is also indicated. In a geometry course, this property is a theorem. So here the theorem is proven using knowledge of the concept of vectors and actions on them.
The teacher can supplement this presentation with his own examples and tasks, but everything that is required for an average level of knowledge in this subject is published here. Moreover, the author left the opportunity for the teacher to dream up and refine what he himself wants in order to create the appropriate atmosphere in the lesson. Don’t forget about the mood for the lesson itself. Then with the help of this presentation you can definitely achieve the desired result.
Definition: The midline of a triangle is the segment connecting the midpoints of its two sides. AK = KS VE = CE KE – midline ABC Definition: the midline of a trapezoid is a segment connecting the midpoints of its lateral sides. A BC K N E AN = NV KE = CE NOT – middle line ABC A B S K E How many middle lines are in the triangle? How many midlines are there in a trapezoid?
Midline of a triangle Theorem. The midline of a triangle is parallel to one of its sides and equal to half of that side. A C B M K Given: ABC, MK – middle line Proof: Since according to the condition MK is the middle line, then AM = MV = ½ AB, SK = KB = ½ BC, Hence, VM AB VC BC 1 2 V – common for ABC and MVK, which means that ABC and MVK are similar according to the second similarity criterion, therefore, VMK = A, which means MC AC. Prove: MK AC, MK = ½ AC MK AC 1 2 From the similarity of the triangles it also follows that, i.e. MK = ½ AC.
Solve the problem F R N ? A B
Proof: Let's carry out A 1 B 1 A B C A1A1 B1B1 O C1C1 According to the condition AA 1, BB 1 are medians, which means BA 1 = CA 1, AB 1 = CB 1, i.e. A 1 B 1 is the middle line. This means A 1 B 1 AB, therefore 1 = 2, 3 = 4. Therefore, triangles AOB and A 1 OB 1 are similar at two angles. This means that their sides are proportional: AO VO AB A1OA1O B1OV1O A1B1A1B1 By the property of the midline of the triangle AB = 2 A 1 B 1, i.e. AO VO AB A1OA1O B1OV1O A1B1A1B1 2 1 Similarly, CO C1OC1O 2 1 We get: C1OC1O AOBOSO A1OA1OV1OV1O 2 1
Midline of a trapezoid Theorem. The midline of the trapezoid is parallel to the bases and equal to their half-sum. A B C K M R Given: ABC - trapezoid MR - midline Prove: MR AK, MR BC MR = Proof: O Let's draw a straight line ME AK through point M, prove that ME will pass through RT. Since ABC is a trapezoid , then BC AK, and, therefore, BC ME AK Since MR is the middle line, then AM = MV, KR = SR E Therefore, MR lies on ME, which means MR AK, MR BC. Let's conduct a VK. According to Thales' theorem, O is the middle of the VC, which means MO is the middle line of the ABC, OR is the middle line of the VSK MR = MO + OR = ½ AK + ½ BC = ½ (AK + BC) = According to Thales' theorem, ME will intersect the SC in the middle of the SC, i.e. at point P.
“Lesson area of a trapezoid” - In a rectangular trapezoid, the base is 5 cm. and 17cm, and the smaller side is 10cm. The teacher sums up the results by asking questions: Who received 5, 4, 3 points? In each case, they formulate a theorem that has been proven. Solving the problem. How to calculate the area of a trapezoid? What elements of plane figures are used in area formulas?
“Problems on the Pythagorean Theorem” - No. 21 Find: X. No. 18 Find: X. No. 27 Find: X. Problems on ready-made drawings (“Pythagorean Theorem”). No. 23 Find: X. No. 25 Find: X. No. 26 Find: X. No. 13 Find: X. No. 20 Find: X. No. 19 Find: X. No. 14 Find: X. You have completed all the proposed tasks. No. 29 Find: X. No. 28 Find: X. No. 30 Find: X. No. 22 Find: X.
"Thales' Theorem" - Thales is widely known as a geometer. Astronomy. Milesian materialist. Let us draw a line EF through point B2, parallel to line A1A3. From the equality of triangles it follows that the sides are B1B2 = B2B3. Thales's theorem. It is believed that Thales was the first to study the movement of the Sun across the celestial sphere. Triangles B2B1F and B2B1E are equal according to the second sign of equality of triangles.
“Theorem of Sines” - The sides of a triangle are proportional to the sines of the opposite angles. Solution: Oral work: Answers to problems based on drawings: Checking homework. Lesson topic: Theorem of sines. Theorem of sines:
“Lesson Pythagorean theorem” - Determine the type of triangle: Introduction to the theorem. Proof of the theorem. Warm up. Pythagorean theorem. And you will find a ladder 125 feet long. Lesson plan: Historical excursion. Show pictures. Solving simple problems. Calculate the height CF of the trapezoid ABCD. Proof. Determine the type of quadrilateral KMNP.
“Pythagorean Theorem 8th grade” - FIGURES. Dividing numbers into even and odd, simple and composite. Given: right triangle a, b legs c - hypotenuse. Height. Bhaskari's proof. Discoveries of the Pythagoreans in mathematics. Given: Right triangle, a, b – legs, c – hypotenuse Prove: c2 = a2 + b2. Smallest side of a right triangle.
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Slide captions:
Middle line (8th grade)
Middle line of the triangle
The middle line of the triangle. Definition: The segment connecting the midpoints of two sides of a triangle is called the MIDDLE LINE OF THE TRIANGLE.
Theorem The middle line of a triangle is parallel to one of its sides and equal to half of that side. i.e.: KM ║ AC KM = ½ AC A B C K M
Solve the problem orally: A B C K M 7 cm Given: M K – avg. line Find: AC?
Work in pairs:
Let's solve the problem: Given: MN – avg. line Find: P ∆ ABC M N A B C 3 4 3.5
Work in pairs:
Midline of trapezoid
Let's remember: A trapezoid is a quadrilateral in which two sides are parallel and the other two sides are not parallel A D B C BC || AD - bases AB łł CD – sides
Midline of trapezoid. Definition: The midline of a trapezoid is the segment connecting the midpoints of its sides. A D B C M N MN – midline of trapezoid ABCD
Theorem about the midline of a trapezoid The midline of a trapezoid is parallel to its bases and equal to their half-sum. i.e.: M N ║ВС║А D М N = ½ (ВС+А 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: Given: AB = 16 cm; CD = 1 8 cm; M N = 15 cm Find: P ABCD = ? M N A D B C
Independent work Task: The middle line of the trapezoid is 5 cm. Find the bases of the trapezoid if it is known that the lower base is 1.5 times larger than the upper base. Solution: A D B C 5 cm Let BC = X cm then AD = 1.5X cm BC+AD = 10 cm X + 1.5X = 10 X = 4 So: BC = 4 cm AD = 6 cm
THANK YOU FOR THE LESSON!!!
The presentation was developed by the mathematics teacher of GBOU secondary school No. 467 of St. Petersburg, Kolpinsky district, Lugvina Natalya Anatolyevna
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