Construct segment A1B1 symmetrical to the segment AB relative to point O. Point O is the center of symmetry. A1. V.O.A. Note: with symmetry around the center, the order of the points has changed (top-bottom, right-left). For example, point A was displayed from bottom to top; it was to the right of point B, and its image, point A1, turned out to be to the left of point B1.
Slide 16 from the presentation "Symmetry of Figures". The size of the archive with the presentation is 680 KB.Geometry 9th grade
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The purpose of the lesson:
- formation of the concept of “symmetrical points”;
- teach children to construct points symmetrical to data;
- learn to construct segments symmetrical to data;
- consolidation of what has been learned (formation of computational skills, division of a multi-digit number by a single-digit number).
On the stand “for the lesson” there are cards:
1. Organizational moment
Greetings.
The teacher draws attention to the stand:
Children, let's start the lesson by planning our work.
Today in mathematics lesson we will take a journey into 3 kingdoms: the kingdom of arithmetic, algebra and geometry. Let's start the lesson with the most important thing for us today, with geometry. I'll tell you a fairy tale, but "A fairy tale is a lie, but there is a hint in it - a lesson for good fellows."
": One philosopher named Buridan had a donkey. Once, leaving for a long time, the philosopher put two identical armfuls of hay in front of the donkey. He placed a bench, and to the left of the bench and to the right of it, at the same distance, he placed completely identical armfuls of hay.
Figure 1 on the board:
The donkey walked from one armful of hay to another, but still did not decide which armful to start with. And, in the end, he died of hunger."
Why didn't the donkey decide which armful of hay to start with?
What can you say about these armfuls of hay?
(The armfuls of hay are exactly the same, they were at the same distance from the bench, which means they are symmetrical).
2. Let's do a little research.
Take a sheet of paper (each child has a sheet of colored paper on their desk), fold it in half. Pierce it with the leg of a compass. Expand.
What did you get? (2 symmetrical points).
How can you be sure they are truly symmetrical? (let's fold the sheet, the dots match)
3. On the desk:
Do you think these points are symmetrical? (No). Why? How can we be sure of this?
Figure 3:
Are these points A and B symmetrical?
How can we prove this?
(Measure the distance from the straight line to the points)
Let's return to our pieces of colored paper.
Measure the distance from the fold line (axis of symmetry) first to one and then to the other point (but first connect them with a segment).
What can you say about these distances?
(The same)
Find the middle of your segment.
Where is it?
(Is the point of intersection of segment AB with the axis of symmetry)
4. Pay attention to the corners, formed as a result of the intersection of segment AB with the axis of symmetry. (We find out with the help of a square, each child works at his own workplace, one studies at the blackboard).
Children's conclusion: segment AB is at right angles to the axis of symmetry.
Without knowing it, we have now discovered a mathematical rule:
If points A and B are symmetrical about a straight line or axis of symmetry, then the segment connecting these points is at a right angle or perpendicular to this straight line. (The word “perpendicular” is written separately on the stand). We say the word “perpendicular” out loud in chorus.
5. Let us pay attention to how this rule is written in our textbook.
Work according to the textbook.
Find symmetrical points relative to the straight line. Will points A and B be symmetrical about this line?
6. Working on new material.
Let's learn how to construct points symmetrical to data relative to a straight line.
The teacher teaches reasoning.
To construct a point symmetrical to point A, you need to move this point from the straight line to the same distance to the right.
7. We will learn to construct segments symmetrical to data relative to a straight line. Work according to the textbook.
Students reason at the board.
8. Oral counting.
This is where we will end our stay in the “Geometry” Kingdom and will do a little mathematical warm-up by visiting the “Arithmetic” Kingdom.
While everyone is working orally, two students are working on individual boards.
A) Perform division with verification:
B) After inserting the required numbers, solve the example and check:
Verbal counting.
- The lifespan of a birch is 250 years, and an oak is 4 times longer. How long does an oak tree live?
- A parrot lives on average 150 years, and an elephant is 3 times less. How many years does an elephant live?
- The bear invited guests to him: a hedgehog, a fox and a squirrel. And as a gift they presented him with a mustard pot, a fork and a spoon. What did the hedgehog give the bear?
We can answer this question if we execute these programs.
- Mustard - 7
- Fork - 8
- Spoon - 6
(The hedgehog gave a spoon)
4) Calculate. Find another example.
- 810: 90
- 360: 60
- 420: 7
- 560: 80
5) Find a pattern and help write down the required number:
3 9 81 2 16
5 10 20 6 24
9. Now let's rest a little.
Let's listen to Beethoven's Moonlight Sonata. A minute of classical music. Students put their heads on the desk, close their eyes, and listen to music.
10. Journey into the kingdom of algebra.
Guess the roots of the equation and check:
Students solve problems on the board and in notebooks. They explain how they guessed it.
11. "Blitz tournament" .
a) Asya bought 5 bagels for a rubles and 2 loaves for b rubles. How much does the entire purchase cost?
Let's check. Let's share our opinions.
12. Summarizing.
So, we have completed our journey into the kingdom of mathematics.
What was the most important thing for you in the lesson?
Who liked our lesson?
It was a pleasure working with you
Thank you for the lesson.
Figures were considered that were symmetrical about a straight line, which was called the axis of symmetry.
In geometry, another type of symmetry is considered, which is called central symmetry or symmetry about a point called center symmetry.
1. Centrally symmetrical points.
If we take some point O, draw a straight line through it and plot equal segments OB and OS on this straight line on opposite sides of point O (Fig. 231), then we get two points B and C, centrally symmetrical relative to point O. Point O is called center symmetry of these points.
Centrally symmetrical with respect to the center O are two points that lie on the same straight line passing through the center O, at equal distances from the center O.
If you rotate the segment OS around point O by 180°, then points C and B will coincide. Two figures are called centrally symmetrical with respect to center O if, when one of them is rotated around this center by 180°, they coincide with all their points.
2. Centrally symmetrical segments.
Let's take two pairs of centrally symmetrical points relative to point O (Fig. 232): OB = OB" and OC = OC". Let's connect points B and C, B" and C" with segments. We obtain segments BC and BC, the ends of which are centrally symmetrical with respect to point O.
If we rotate the drawing around point O by 180°, then points B" and C" will take the position of points B and C, respectively. Segments B "C" and BC will align, they are centrally symmetrical. Centrally symmetrical segments are equal.
3. Centrally symmetrical triangles.
Let's take three pairs of centrally symmetrical points relative to some point O (Fig. 233):
OA = OA", OB = OB" and OS = OS.
By connecting point A with points B and C, and point A" with points B" and C", we obtain two triangles. These triangles are centrally symmetrical with respect to point O, which is the center of symmetry.
When the drawing is rotated around point O by 180°, points A", C" and B" will take the positions of points A, C and B, respectively, i.e. /\ A"C"B" and /\ ASV will be combined. Centrally symmetrical triangles are congruent. Any symmetrical figures are equal in the same way.
4. Symmetry of a parallelogram.
Big number figures has the property that when the drawing plane is rotated 180° around a certain point, the new position of the figure coincides with the original one. Such figures are called centrally symmetrical. A parallelogram is one of these figures; it is centrally symmetrical with respect to the point of intersection of its diagonals (Fig. 234).
In fact, since OS = OB and OA = OD, then points C and B, as well as A and D, are symmetrical with respect to the center O. If the parallelogram is rotated 180° around the intersection point of its diagonals, then the new position of the parallelogram will coincide with the original one.
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Axial and central symmetry are used by almost all graphics programs when displaying images horizontally and vertically (axial symmetry) and rotating them by 180° (central symmetry).
1. Construct a parallelogram in any graphics program (Paint, PhotoShop, etc.) using the central symmetry method.
2. Copy the drawing into the Paint program and find the center of symmetry of the triangles.