Click one: What is axial symmetry? What is an axisymmetric figure? What's the difference between them?

If a graph is folded along a straight line, if it can overlap with another graph, the two graphs are said to be symmetrical about this straight line, which is called symmetry axis, and the point where they overlap after folding is the corresponding point, which is called symmetry point. The symmetry of two graphs about a straight line is also called the symmetry axis.

If a figure is folded along a straight line, the parts on both sides of the straight line can overlap each other. This figure is called an axisymmetric figure, and this straight line is its axis of symmetry.

Axisymmetry is the relationship between two graphs, and an axisymmetric graph is the feature of a graph.

Click 2: What are the properties of the axis symmetry of a graph?

The symmetry of graphs has the following two properties: if two graphs form an axis symmetry, then the axis of symmetry is the perpendicular to the line segment connected by any pair of corresponding points; The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points. (2) Axisymmetry refers to the shape and position relationship between two figures, and the two figures with axial symmetry are conformal; Axisymmetric graph is a kind of graph with special shape. An axisymmetric figure is divided into two figures along the axis of symmetry, which are congruent and symmetrical.

Click 3: What are the properties of the midline of the line segment?

The distance between the point on the vertical line of the line segment and the two end points of the line segment is equal; On the contrary, the points with equal distance from the two ends of a line segment are on the midline of the line segment, so the midline of the line segment can be regarded as the set of all points with equal distance from the two ends of the line segment.

Click Four: Symmetric Transformation Properties and the Law of Coordinate Symmetry

Properties of axisymmetric transformation;

The shape and size of the graph obtained by (1) axisymmetric transformation are exactly the same as the original graph.

(2) Every point on the graph obtained by axisymmetric transformation is a symmetrical point of a point on the original graph about the axis of symmetry.

(3) The line segment connecting any pair of corresponding points is vertically bisected by the axis of symmetry.

The coordinate of the point P(x, y) about the axis symmetry of X is (x,-y);

The coordinates of the point where the point P(x, y) is symmetrical about y are (-x, y);

The coordinates of the point P(x, y) symmetrical about the origin are (-x, -y).

The coordinate of the point where point P(x, y) is symmetrical about 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).

One of these types:

Example 1: As shown in the figure, it is known that the straight line MN△ABC is calculated as △ a1b1,then △A 1B 1C 1 and △ABC are symmetric about MN.

First, determine the symmetry point of the key point about the straight line. According to the concept of axial symmetry, the straight lines MN in the three directions of A, B and C are crossed into vertical lines respectively, and the vertical lines are extended twice to connect these three points, so that the symmetrical points of points A, B and C about the straight line MN can be obtained.

answer

Practice: (1) as AD⊥MN in D, extend AD to A 1 so that A 1d = AD, and get the symmetry point A 1

(2) In the same way, make the symmetry points of point B and point C about MN B 1 and C 1.

(3) connect A 1, B 1 and C 1 in turn.

∴△A 1B 1C 1 is what you want.

Type 2:

Example 2: As shown in the picture, the shepherd boy is herding cattle at A, and his home is at B. The distances from A and B to the river bank are AC and BD respectively.

And AC = BD, if the distance from A to the midpoint of CD on the river bank is 500cm .. Q:

(1) The shepherd boy led A's cow to the river to drink water before going home. Where is the shortcut to drinking water?

(2) What is the shortest distance?

The analysis of the problem can be transformed into taking a little m on the CD to minimize its AM+BM; On the above basis, practical problems are transformed into mathematical problems by using the properties of triangles. If A and B are on both sides of a straight line, it is natural to think of connecting AB, and the intersection point is the point to be found. But in this question, A and B are on the same side of the straight line, how can they be transformed into different sides? We tend to understand "folding" as "axial symmetry". If point A is the symmetrical point A 1 about a straight line, then the distance from any point on the straight line to A and A 1 is always equal.

The answer can be transformed into two points A and B on the same side of a known straight line CD.

Do a little m minimization AM+BM on CD,

Let's talk about the symmetry point of a point about CD A 1.

Then connect A 1B, cross the CD at point m,

Then point m is the desired point.

Proof: (1) Take any point M 1 on the CD and connect A 1M 1, AM 1, BM 1, AM.

∵ The straight line CD is the symmetry axis of A and A 1, and M and M 1 are on the CD.

∴am=a 1m,am 1=a 1m 1

∴AM+BM=AM 1+BM=A 1B

In △A 1 M 1B

∫ a1m1+BM1> am+bn means that AM+BM is the smallest.

(2) From (1), we can get AM = AM 1, A 1C = AC = BD.

∴△A 1CM≌△BDM

∴A 1M=BM,CM=DM

That is, m is the midpoint of CD, and a 1b = 2am.

AM = 500m.

∴ shortest distance a1b = am+BM = 2am =1000m.

Type 3:

Example 3: There is a point P in the acute angle ∠AOB. Try to determine two points C and D on OA and OB to make the circumference of △PCD shortest.

It is analyzed that the circumference of △PCD is equal to PC+CD+PD. In order to make the circumference of △PCD shortest, according to the shortest line segment between two points, the size of PC+CD+PD only needs to be equal to the distance between some two points, so considering the symmetrical points E and F of point P about straight lines OA and OB, the circumference of △PCD is equal to the length of line segment EF.

explain

Practice: as shown in the figure. (1) Make the symmetry point E of point P about straight line OA;

② Make the symmetrical point f of point P about straight line OB;

③ Connect EF, OA and OB at points C and D respectively, then points C and D are required points.

Proof: connect PC and PD, then PC=EC, PD = FD.

Take any H point in OA that is different from point C and connect HE, HP and HD, then He = HP.

∫ perimeter PHD

= HP+HD+PD = HE+HD+DF & gt； ED+DF=EF

And the circumference of delta △PCD.

=PC+CD+PD=EC+CD+DF=EF

∴△PCD has the shortest circumference.

1. As shown in the figure, △ABC and △AED are symmetrical about the straight line 1. If AB=2cm, ∠ C = 95, AE = _ _ _ _ _ _ ∠ D = _ _ _ degrees.

By analyzing the symmetry of two triangles about a straight line, we can get AE=AB, ∠C=∠D, and then we can get the answer.

The answer is 2 cm; 95

2. It is known that, as shown in the figure, △ABC is an equilateral triangle, extending from BC to D, extending from BA to E, making AE = BD and connecting CE and DE. Verification: CE = DE.

The key to solving analytical problems is to make correct auxiliary lines. To prove CE = DE, it is enough to prove that E is on the vertical line of CD. Therefore, we construct an axisymmetric figure about perpendicular bisector of CD to prove it.

explain

Proof: extend BD to F, make DF = BC, and connect EF.

AE = BD, △ABC is an equilateral triangle.

∴BF=BE，∠B=

△ BEF is an equilateral triangle.

∴△BEC≌△FED

∴CE=DE

3. As shown in the figure, point D is the intersection of the bisector of ∠BAC in △ABC and the median line de of BC, DG⊥AB is at point G, and the extension line of DH⊥AC is at point H, which proves BG = ch.

By analyzing the AD bisector ∠BAC and DG⊥AB and DH⊥AC, we can get DG=DH (the nature of the angular bisector), while DE is the median line of BC, and BD=CD can be obtained from the nature of the median line, so we can use "HL" to prove Rt△BDG≌Rt△CDH and get BG =.

explain

It is proved that the point D connecting BD and CD is on the bisector of ∠BAC, DG⊥AB and DH ⊥ AC.

∴DG=DH (the distance from the point on the bisector of the angle is equal to both sides of the angle)

De is perpendicular bisector from BC.

∴DB=DC (the point on the middle vertical line of the line segment is equal to the distance between the two endpoints of the line segment).

∵DG⊥AB、DH⊥AC ∴∠BGD=∠CHD=90

At Rt△BDG and Rt△CDH,

∴Rt△BDG≌Rt△CDH(HL) ∴BG=CH (the corresponding sides of congruent triangles are equal).

1. As shown in the figure, it is a pattern folded with paper, and there is an axisymmetric pattern ().

1。

According to the characteristics of axisymmetric graphics, envelopes, planes and trousers can be completely overlapped by folding, while some jackets can be overlapped and some can not be overlapped after folding.

Answer c

2. The coordinate of the point M (-2, 1) with respect to the axis of X is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

The coordinates of a point about the axis symmetry of X remain unchanged, and the coordinates on the Y axis become the original reciprocal. The line between two symmetrical points is perpendicular to the axis symmetry.

Solution (-2,-1); Perpendicular to each other.

3. It is known that two nonparallel line segments AB and A ′ b ′ are symmetrical about the straight line 1, and the straight line where AB and A ′ b ′ are located intersects with point P, and the following four conclusions are drawn: ① AB = A ′ b ′; ② Point P is on the straight line 1; ③ If a and A' are corresponding points, the straight line 1 vertically bisects the line segment aa'; (4) If both B and B‘' are corresponding points, then PB = PB', and the correct one is ().

A.①③④ B.③④ C.①② D.①②③④

According to analysis, two figures are symmetrical about a straight line and have five characteristics: 1. The corresponding line segments are equal, 2. The extension lines of the corresponding line segments intersect at one point and are on the axis of symmetry 3. Symmetry axis bisects the connecting line of corresponding points vertically, 4. The distance from the intersection point to the corresponding point is equal, thus judging four points.

Answer d

4. There will be some interesting symmetrical forms in the operation of numbers. Fill in the blanks according to the form of Equation ①, and check whether the equation holds.

① 12×23 1= 132×2 1;

② 12×462=___________;

③ 18×89 1=__________;

④24×23 1=___________.

Analysis of 264× 21; 198×8 1; 132×42 .

According to the characteristics of axial symmetry, numbers are also symmetrical, which is modeled after symmetry.

As shown in the figure, please write down the coordinates of each vertex in △ABC. Draw a straight line m: x =- 1 in the same coordinate system as △ a ′ b ′ c ′, where △ABC is symmetrical about the straight line m. If P(a, b) is a point on the AC side of △ABC, please indicate its corresponding point in △ a ′ b ′ c ′.

Analytic straight line M: x =- 1 means that the abscissa of any point on straight line M is equal to-1, so the intersection point (-1, 0) is the parallel line of Y axis, that is, straight line M. After drawing straight line M, make symmetrical points A' and C' of point A and point C about straight line M, and point B is in straight line M.

explain

The coordinates of each vertex in (1)△ABC are A (1, 4), B (- 1, 1) and C(2 1) respectively.

(2) As shown in the right figure, take the intersection point (-1, 0) as the parallel line m of the Y axis, that is, the straight line X =- 1.

A'(-3) As shown in the right figure, make points A' (-3,4), B'(- 1, 1), C'(-4, 1) which are symmetrical about the straight line M, and connect A', B' and C in turn.

(4) It is observed that the vertical coordinates of the three groups of symmetrical points have not changed, and the horizontal coordinates can all be expressed as 2× (- 1) minus the horizontal coordinates of the corresponding points, so the coordinates of the corresponding points of point P are (-2-a, b).

Class assignments:

first-rate

1. The axis of symmetry of an axisymmetric graph is _ _ _ _ _ _ _ _.

2. If one internal angle of an isosceles triangle is 0, then the other two internal angles are _ _ _ _ _ _ _ _.

3. Write the English letters of six axisymmetric figures: _ _ _ _ _ _ _ _ _ _ _ _ _ _.

4. Write five axisymmetric Chinese characters: _ _ _ _ _ _.

5. The isosceles triangle has _ _ _ _ _ _ symmetry axes; The five-pointed star has _ _ _ _ _ _ symmetry axes; The symmetry axis of the angle is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

6. The following statement is wrong ()

A. Two triangles symmetrical about a straight line must be congruent.

B. an axisymmetric figure has at least one axis of symmetry.

C. congruent triangles must be symmetrical about a straight line.

D. an angle is a figure that is symmetrical about its bisector.

7. In the figure below, the axisymmetric figure is ().

8. In the figure below, it is an axisymmetric figure with two axes of symmetry ().

9. If a point P satisfies PA=PB=PC within the acute angle △ABC, then this point P is △ABC ().

A. Intersection of three bisectors of angles B. Intersection of three median lines

C. Intersection of three heights D. Intersection of three sides of perpendicular bisector

10. In △ AC>BC, the perpendicular line of the side AC>BC and AB intersects with AC at point D. Given AC=5, BC=4, the circumference of △BCD is ().

a9 b . 8 c . 7d . 6

b grade

1 1. The symmetry axis of two points that are not coincident on the plane is _ _ _ _ _ _ _ _, and the line segment is an axisymmetric figure with _ _ _ _ _ _ _ _ _.

12. The circumference of an isosceles triangle with two sides of 4cm and 8cm is _ _ _ _ _ _ _ _ cm.

13. Name at least three things that are axially symmetrical in life: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

14. If AC is the height of isosceles ABC, then AC is also _ _ _ _ _ _ _ _ _, or _ _ _ _ _ _ _ _.

15. The circumference of an equilateral triangle is 30 cm and the height of one side is 8 cm, so the area of the triangle is _ _ _ _ _ _ _.

16. In the following figures, the one that is not axisymmetric is ().

A. equilateral triangle B. isosceles right triangle

C. Unequal triangle D. Line segment

17. Of the following statements, the correct one is ().

A. Two triangles symmetrical about a straight line are congruent triangles.

B. congruent triangles is symmetrical about a straight line

C if two figures are symmetrical about a straight line, they must be located on both sides of the straight line.

D. Two congruent triangles with a common edge are symmetrical about the line where the common edge lies.

18. In the four graphs of line segment, two intersecting lines, isosceles triangle and circle, the number of axisymmetric graphs is ().

A. 1 B.2 C.4 D.3

19. As shown in the figure, DE is the middle vertical line of the AC side in ABC. If BC=8 cm and AB= 10 cm, the circumference of ABC is () cm.

16

20. In the following figure, the figure that is not axisymmetric is ().

A. a triangle with two equal sides and an angle.

C. an isosceles triangle with one angle. A triangle with an inner angle of.

Class c

2 1. Among the symbols shown in the figure, () is an axisymmetric figure.

1。

22. The following plane graphics, not axisymmetric graphics is ().

23. As shown in the figure, among the following four graphs, the graph with the largest number of symmetry axes is ().

24 As shown in the figure, in △ABC, AB=AC= 14cm, D is the midpoint of AB, DE⊥AB intersects AC at D and E, and the circumference of △EBC is 24cm, then BC = _ _ _ _ _ _ _

25. Two non-parallel line segments AB are symmetrical about a straight line. AB intersects the straight line at point P, and the following conclusions are drawn: ① AB =; ② Point P is on a straight line; (3) If point A and point A are symmetrical points, the line segments are bisected vertically; (4) If point B is a symmetrical point, PB=, where the correct one is (only fill in the serial number).

26. When the paper money with numbers written on it is placed vertically with the mirror (as shown):

The following is a string of numbers seen in the mirror, actually.

27. This picture is a figure composed of three small squares. Please draw a small square in the picture, so that the drawn picture is axisymmetric.

28. Find out the geometric figures of one symmetry axis, two symmetry axes, three symmetry axes and four symmetry axes respectively, and draw them (including symmetry axes).

29. As shown in the figure, △ABC and △ are symmetrical about the straight line m. 。

(1) Point out the symmetry point by combining the graph.

⑵ What is the relationship between connecting A, A and straight line M and line segment?

(3) What is the relationship between extension line segments AC and their intersections with straight lines M? What about the intersection of other corresponding line segments (or their extension lines)? What patterns have you found? Please describe it and communicate with your peers.

30. The internal angle of an isosceles triangle is less than twice that of another triangle. Find the degrees of the three internal angles of this triangle. (Consider two situations)

A level answer

1. straight line

2.35 degrees, 35 degrees

3.ACDEHI

4. Big, sky, mouth, sky, goods and fields

5. 1, 5, bisector

6.C

7.D

8.D

9.D

10.A

B-level answer

1 1. perpendicular bisector of the line segment obtained by connecting two points.

12.20cm

13. Temple of Heaven, blackboard, etc.

14. Center line, vertex bisector

15.40

16.C

17.A

18.C

19.D

20.B

C level answer

2 1.C

22.A

23.B

24.10cm

25.①②③④

26.526778022

27. The parts on both sides of the crease are symmetrical about the crease. Please refer to the following figure:

28. As shown in the figure:

29.( 1) omitted; (2) M is divided into AA/; (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.

30. The first situation: 52.5 degrees, 52.5 degrees, 75 degrees; The second situation: 48 degrees, 66 degrees, 66 degrees