Two kinds of differences
chart
Learning point
axial symmetric figure
(mnemonic method: one word and one picture)
Axial symmetry
(mnemonic method: two numbers are missing from the word)
The confusion of mastery and memory and the learning difficulties that need to be broken through
What are the key differences among (1), "axisymmetric figure" and "axisymmetric"? Which is "one number" and which is "two numbers"? In the process of study and review, students are always confused with each other and can't remember. Where is the mystery?
(2) "Complete coincidence" and "complete identity" are two different concepts. Completely coincident figures can be completely identical, but completely identical figures are not necessarily completely coincident.
This is easy to be confused in learning.
In order to review memory conveniently, the author's original memory method.
Interpretation of the meaning of memory formula;
Judging by the number of words of "axisymmetric figure" and "axisymmetric". We might as well count that the word "axisymmetric figure" has five words and the word "axisymmetric" has three words. Comparing the words of two nouns, "axisymmetric figure" has two more words than "axisymmetric". So we come to the conclusion that there are many words "axisymmetric figure".
"Axisymmetric" is a rare word. So we use reverse thinking to judge and remember the number of numbers under these two nouns. The way to remember is: there is only one number for many words and two numbers for few words. Based on this, we extracted the formula of memory:
Formula: one word has one more number and one word has two fewer numbers.
Reverse thinking memory method: the word "axisymmetric figure" has one more figure.
The word "axial symmetry" is missing (yes) two numbers.
Remember this formula. When studying this kind of material or doing axisymmetric problems, you will read this formula quietly and look at the following series of complicated contents, which will produce an overwhelming pleasure.
definition
Definition of Multi-Axisymmetric Graph (Only One Graph)
Axisymmetric figure refers to a figure. If you fold it in half along a straight line and the two folded parts can overlap each other, then this figure is called an axisymmetric figure.
Remarks:
1. Axisymmetric graphics are folded along a straight line, and the parts on both sides of the straight line overlap each other. There are two elements: one is to fold along a straight line, and the other is that the two parts overlap each other.
② According to the definition of axisymmetric figure, we can know that axisymmetric figure has two important properties: ① The axis of symmetry is vertical and bisects the line segment connecting two symmetrical points. ② Two axisymmetric figures are congruent. However, it should be noted that axisymmetric figures are two congruences in special relative positions, so congruences are not necessarily axisymmetric figures.
Definition of axisymmetry without words (with two pictures)
Axisymmetry refers to the relationship between two figures. If one of the figures is folded along a straight line and can overlap with the other figure, the two figures are said to be symmetrical about this straight line, or they form an axis symmetry.
Remarks:
About two figures that are symmetrical about a straight line, the corresponding line segments are equal and the corresponding angles are equal.
Brief definition
Two parts in a graph are symmetrical about a straight line.
Two figures are symmetrical about a straight line.
Definition hint
① Axisymmetric graphics are graphics with special characteristics, which can completely overlap after being folded in half, that is, parts on both sides of the axis of symmetry are congruent.
② Axisymmetric graphics may have more than one axis of symmetry.
(1), there are two graphics, can completely overlap, the shape and size are exactly the same.
② Two figures can overlap after being folded in half along the symmetry axis.
③ There is only one symmetry axis between two figures.
axis of symmetry
This straight line is the symmetry axis of this figure.
The axis of symmetry is a straight line, not a ray or a line segment.
② Axisymmetric graphics have only one axis of symmetry, but multiple axes of symmetry.
This straight line is the symmetry axis of these two figures.
The axis of symmetry is a straight line, not a ray or a line segment.
② Generally, two symmetrical figures have only one axis of symmetry.
symmetry point
For a graph, the points when it is folded along this straight line and overlapped with each other are called symmetrical points (also called corresponding points).
For two graphs, the point where the two graphs overlap each other after folding is called a symmetrical point (also called a corresponding point).
axisymmetric
This figure is symmetrical about this straight line (axis).
These two figures are symmetrical about this straight line (axis).
Axisymmetric transformation
(1) and the definition of "axisymmetric transformation"
The process of getting its axisymmetric figure from a plane figure is called axisymmetric transformation.
② Axisymmetric transformation is a moving process.
Axisymmetric transformation is a kind of transformation, which talks about the process of getting a figure that is axisymmetric with it from a figure, and it is a moving process.
(3) Conversion between axisymmetric graphics and axisymmetric graphics.
Transformation of axisymmetric figure: an axisymmetric figure can also be regarded as a part of it, which is expanded by axisymmetric transformation.
Axisymmetric transformation: either of the two axisymmetric figures can be regarded as the result of the axisymmetric transformation of the other figure.
axial symmetric figure
Axial symmetry
chart
If the color is not considered in the figure below, the pattern shown is an axisymmetric figure, and the straight line L is its axis of symmetry.
Determine which of the listed graphs are axisymmetric. Is there only a fifth one that isn't?
Question explanation:
1, Q: Are angles with different lengths on both sides axisymmetric?
A: Yes, its symmetry axis is the straight line where its angular bisector lies. Because the definition of an angle is: the figure surrounded by two rays emitted from a point is called an angle. Because the ray is infinitely extended, even if the lengths of the two sides are different, it is still an axisymmetric figure.
Axisymmetric property theorem
(Axisymmetric property theorem is also the three properties of axisymmetric figure and axisymmetric, referred to as "axisymmetric property" for short)
Axisymmetric property theorem ①. Two graphs that are symmetrical about a straight line are conformal. (It can be expressed as the congruence of two symmetrical graphs)
This theorem provides a basis for "proving the congruence of two graphs".
Axisymmetric property theorem ②. If two figures (about a straight line) are symmetrical, then the symmetrical axis is the middle vertical line connecting the symmetrical points.
This theorem provides a basis for proving that "a straight line is the midline of a line segment".
Axisymmetric property theorem ③. Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
This theorem provides a basis for proving that "three straight lines intersect at one point".
Remarks:
(1), congruent graphs are not necessarily axisymmetric, and axisymmetric graphs must be congruent.
② The nature of axial symmetry is one of the bases to prove that the line segments are equal, the line segments are vertical and the angles are equal. For example, if two figures are known to be symmetrical about a straight line, their corresponding edges are equal and their corresponding angles are equal.
axisymmetric
Decision theorem
(This theorem is also the inverse theorem of Axisymmetric Property Theorem ③) If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.
This theorem provides a method to judge whether two graphs are symmetrical about a straight line.
trait
Characteristics of axisymmetric graphics;
Axisymmetric graphics are the characteristics of graphics themselves.
Its characteristic is that it can be folded along a straight line, and the parts on both sides of the straight line can overlap each other.
Characteristics of two symmetrical graphs:
Axisymmetry is the relationship between two figures.
The characteristic of two symmetrical figures is that they turn 180 degrees along the symmetry axis and overlap, and the distance between the corresponding points and the symmetry axis is equal.
differentiate
(difference)
Axisymmetric graphics are just a kind of graphics with special shapes.
Axisymmetry is the positional relationship between two figures.
There is not necessarily only one axis of symmetry.
There must be only one axis of symmetry,
Symmetry points are on the same graph.
The symmetry points are on two graphs respectively,
get in touch with
(Similarities)
Axisymmetric graphics are folded in half along the axis of symmetry, and the two parts of the graphics overlap.
The symmetry axis is folded along the symmetry axis, and the two figures overlap.
If an axisymmetric figure is divided into two parts along the axis of symmetry, then the two figures are axisymmetric about this straight line.
If two axisymmetric figures are regarded as a whole, then it is an axisymmetric figure;
Note: The similarity between them is that they can overlap each other after being folded along a straight line. However, the axisymmetric figure is folded in half along the axis of symmetry, and the two parts of the figure overlap. The symmetry axis is folded along the symmetry axis, and the two figures overlap.
Method of identifying symmetry axis
The identification method of symmetry axis is the aforementioned "Axisymmetric Judgment Theorem". The function of this theorem is to judge whether two figures are symmetrical about a straight line, which is the main basis for making symmetrical figures.
Method of finding and drawing symmetry axis
1. Find any set of symmetry points of an axisymmetric graph.
2. Connect the symmetrical points.
3. Draw the middle vertical line of the line segment connected by the symmetry point, which is the symmetry axis of the graph.
Remarks: No matter whether two figures are symmetrical or the axis of symmetry of an axisymmetric figure, it is the middle perpendicular of the line segment connected by any pair of corresponding points. Therefore, as long as we find any pair of corresponding points and make the median perpendicular of the connected line segments, we can get their symmetry axes.
Methods and steps of making axisymmetric graphics
1, the steps of drawing a symmetric point of a known special point:
(1), mark the vertical line with the known point A as the known symmetry axis, and mark the vertical foot O. ..
(2) On the other side of this straight line, starting from the vertical foot O, cut the line segment OA equal to the distance from the known point A to the vertical foot O, then the cut-off point A' is the symmetrical point of the point A about the straight line of the symmetry axis.
2. The step of drawing a symmetrical figure of a known figure: find out the points on the figure according to the coordinates of the corresponding points and connect them.
① Finding known points: determine some special points in the graph.
(2) Draw corresponding points: find the symmetrical points of known points about known symmetry axes.
③ Connecting line: connecting symmetrical points. Connect these symmetrical points in turn to form a symmetrical figure that meets the requirements.
Remarks: To draw a symmetrical figure of a known figure, we must first make clear the following properties of axial symmetry:
This picture is a symmetrical figure of a known figure, and its axis of symmetry is a straight line.
② A straight line that is vertical and bisects a line segment is called the median line of this line segment, or median line. The distance between the point on the vertical line in the line segment and both ends of the line segment is equal.
③ In an axisymmetric figure, the distances between the corresponding points on both sides of the axis of symmetry are equal.
④ In an axisymmetric figure, the axis of symmetry divides the figure into two equal parts.
⑤ If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.
Note that in the graphic topic of symmetry axis, if a triangle is inclined on the vertical line as the symmetry axis, it is necessary to break the habitual thinking that the origin is generally on the left side of the symmetry axis and the corresponding point is generally on the right side of the symmetry axis, and flexibly stagger the letter positions of the corresponding points on both sides of the symmetry axis.
3. Draw a symmetrical figure of a known circle.
If the topic draws an axisymmetric figure of a known circle about a straight line. There is a known circle, and its center coordinates A(a, b) and radius r can be known. As a straight line passing through the center point, the vertical line intersects with C, and the straight line extends to intersect with point B, so that AC=BC, with point B as the center and point R as the radius, is the desired figure.
4. Given one half of an axisymmetric figure and the axis of symmetry, how do you find the other half?
Draw a vertical line from each key point to the symmetry axis and extend the same unit to get the corresponding points of each point, and then connect them in turn. Let AO⊥L be at point O, and extend it, and intercept OA ′ = OA on the extension line to get the symmetrical point of point A.
A', in the same way, connect the symmetry points of the other key points in the left picture about the straight line L in the order in the left picture.
Correctness audit
How to check whether the axisymmetric figure you drew is correct? Just look at whether the distance from the corresponding point to the symmetry axis is equal.
Expressed in coordinates
Axial symmetry
(1) For the method of finding the corresponding point, please refer to the following four points. (2) Look at the examples in the courseware below to deepen the understanding of the above provisions. Examples of free courseware are the following courseware of Baidu Outlets: 12.2.2 Axisymmetric courseware with coordinate 2.
(1), about the axis symmetry (the following example is best to draw a picture on paper, which is clear at a glance. )
The coordinates of the point P(x, y) which is axisymmetrical about X are (x, -y).
The coordinates of the point where the point P(x, y) is symmetrical about y are (-x, y).
② Symmetry about the origin
The coordinates of the point where the point P(x, y) is symmetrical about the origin are (-x, -y).
③ Symmetry of the straight line parallel to the coordinate axis
The coordinate of the point where the point P(x, y) is symmetrical about the straight line x=m is (2m-x, y).
The point P(x, y) is symmetrical about the straight line y=n, and the coordinate of this point is (x, 2n-y).
④ Symmetry of bisector of coordinate axis.
The point P(x, y) is symmetrical with respect to the bisector y=x of the first and third quadrant coordinate axes, and the coordinate of the point is (y, x).
The point P(x, y) is symmetrical about the bisector y= -x of the second and fourth quadrant coordinate axes, and the coordinate of this point is (-y, -x).
Examples of judging axisymmetric figures
1. In addition, by studying the definition of symmetry axis, we can know that "symmetry axis is a straight line", so it is necessary and correct to emphasize the words "straight line".
2. Because "the middle vertical line of the line segment" can be simply called "the middle vertical line", the "middle vertical line" is used in this paper.
3. When judging, we should especially remember the following concepts.
(1), according to the definition to distinguish which graphics are axisymmetric graphics and which are centrally symmetric graphics. The method of distinguishing is simply:
What can be folded along the central axis is an axisymmetric figure.
Rotate 180 degrees to make it coincide with the center symmetry.
2. The definition of an axisymmetric figure is that if a figure is folded in half along a straight line, the figures on both sides can completely overlap, and this figure is called an axisymmetric figure.
③ The symmetry axis is a straight line! ④ A straight line that is vertical and bisects a line segment is called the median line of this line segment, or median line. The point on the vertical line in the line segment is equal to the distance between the two ends of the line segment. ⑤ In an axisymmetric figure, the distances between the corresponding points on both sides of the axis of symmetry are equal. 6. Axisymmetric graphic congruence. If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.
Common graphic symmetry comparison table
(Note: If there is a symmetrical center in the table below, it means that the graph is both an "axisymmetric graph" and a "centrosymmetric graph". )
name
axis of symmetry
amount
Where is the axis of symmetry?
Axisymmetric concurrency
Central symmetry
Yes or no
symcenter
straight line
Countless articles
One is itself, and the others are perpendicular to any point on the straight line.
breakdown
not have
ray
1
Is the straight line where the light is.
breakdown
not have
line segment
2
(1), where the line segment is located; (2) The median vertical line of the line segment.
Central symmetric graph
Midpoint of line segment
corner
1
The straight line where the angular bisector lies.
breakdown
not have
circle
Countless articles
The straight line where the diameter of a circle lies.
Central symmetric graph
centre of a circle
isosceles triangle
1
The position of the bisector of the top corner (or the midline of the bottom edge and the height on the bottom edge).
straight line
breakdown
not have
equilateral triangle
three
Three bisectors of the top corner (or the center line of the bottom edge and the height on the bottom edge)
In a straight line
breakdown
not have
parallelogram
It is centrosymmetric, but not axisymmetrical.
Central symmetric graph
Diagonal crossing
rectangle
2
Two sets of opposite median vertical lines.
Central symmetric graph
Diagonal crossing
diamond
2
Two groups of straight lines connected by diagonal vertices
Central symmetric graph
Diagonal crossing
square
four
Perpendicular bisector of two opposite sides and a straight line connected by two diagonal vertices
Central symmetric graph
Diagonal crossing
isosceles trapezoid
1
Perpendicular bisector with two bottoms.
breakdown
not have
Regular even polygon
Central symmetric graph
Diagonal crossing
Regular odd polygon
breakdown
not have
26 letters
A \ b \ c \ d \ e \ h \ I \ k \ m \ o \ t \ u \ v \ w \ x \ y * *16 axisymmetric graph.
A \ h \ I \ m \ o \ t \ u \ v \ w \ x \ y * * *1left and right symmetry. B\C\D\E\K\ The other five are symmetrical up and down. O\H\X is both an axisymmetric figure and a centrally symmetric figure.
In order to make it easier for readers to understand, the chart is further described as follows:
① The line segment is an axisymmetric figure.
It has two axes of symmetry.
One is the straight line where the line segment is located, and the other is its median vertical line.
② The straight line is an axisymmetric figure.
It has countless symmetry axes.
One is the straight line itself, and the other is any vertical line of the straight line (please pay special attention that it is not the middle vertical line, and the vertical line and the middle vertical line are completely different. Compared with the four figures of straight line, angle, line segment and equilateral triangle, the most symmetrical line is straight line. )
③ The angle is an axisymmetric figure.
It has only 1 symmetry axes.
The straight line where the angular bisector lies is its axis of symmetry.
④ The isosceles triangle is an axisymmetric figure.
It has only 1 symmetry axes.
The vertical centerline of the base is its axis of symmetry.
⑤ An equilateral triangle is an axisymmetric figure.
Because it is equilateral, it has three axes of symmetry.
The perpendicular of each side is its axis of symmetry.
6. This circle is an axisymmetric figure.
It has countless symmetry axes.
A straight line with the same diameter as a circle is the symmetry axis of the circle, and any diameter of the circle is its symmetry axis.
(Note: When two equal circles are symmetrical, there is only one symmetry axis) ⑦. The two straight lines where the midpoint of the opposite side of the rectangle is located are the two symmetrical axes of the rectangle;
8 The two straight lines where the midpoint of the opposite side of a square is located and the two straight lines where the diagonal is located are its four symmetry axes.