①CE = CF； ②? AEB=75？ ; ③BE+DF = EF； ④S squared ABCD=2+.
The correct serial number is 1224 (fill in everything you think is correct).
Test center: the nature of the square; Congruent triangles's judgment and nature; Properties of equilateral triangles.
Analysis: According to the knowledge of triangle congruence, we can judge whether ① is right or wrong; According to the quantitative relationship between the angles, the sum of the internal angles of the triangle is 180? Judge whether ② is right or wrong; The correctness of ③ can be judged according to the knowledge of the middle vertical line, and ④ can be judged by solving the triangle and finding the square area.
Solution: solution: ∵ quadrilateral ABCD is a square,
? AB=AD，
∵△AEF is an equilateral triangle,
? AE=AF，
In Rt△ABE and Rt△ADF,
,
? Rt△ABE≌Rt△ADF(HL)，
? BE=DF，
∫ BC =DC,
? BC﹣BE=CD﹣DF，
? CE=CF，
? (1) The statement is correct;
CE = CF，
? △ECF is an isosceles right triangle,
CEF=45？ ,
∵? AEF=60？ ,
AEB=75？ ,
? 2 The statement is correct;
As shown in the figure, connect AC, cross EF at G point,
? AC？ EF, AC divides EF equally,
∵? Kadav,
? DF？ FG，
? BE+DF？ EF，
? 3 wrong statement;
EF = 2，
? CE=CF=，
Let the side length of a square be a,
In Rt△ADF,
a2+(a﹣ )2=4，
Solve a=,
Then a2=2+,
The square of s ABCD=2+,
This statement is correct.
So the answer is 1224.
Comments: This question mainly examines the knowledge points of square nature. The key to solve this problem is to master the correct method of congruent triangles's proof and auxiliary line. This question is not difficult, but it is a bit troublesome.
(20 13? Huanggang) It is known that △ABC is an equilateral triangle and BD is the center line. If BC extends to e, CE=CD= 1, and DE is connected, then DE=.
Test center: the nature of equilateral triangle; On the judgement and properties of isosceles triangle+120886.18888888888888
Analysis: According to the properties of isosceles triangle and triangle outer angle, BD=DE and BC are obtained. In Rt△△BDC, BD can be obtained by Pythagorean theorem.
Solution: Solution: ∫△ABC is an equilateral triangle,
ABC=？ ACB=60？ ，AB=BC，
∫BD is the centerline,
DBC=？ ABC=30？ ,
CD = CE，
E=？ CDE，
∵? E+？ CDE=？ ACB，
E=30？ =? DBC，
? BD=DE，
∫BD is AC midline, CD= 1,
? AD=DC= 1，
∫△ABC is an equilateral triangle,
? BC=AC= 1+ 1=2，BD？ Communication,
In Rt△△BDC, from Pythagorean theorem, BD= =,
That is, DE=BD=,
So the answer is:
Comments: This topic examines the properties of equilateral triangle, Pythagorean theorem, isosceles triangle, the application of triangle outer angle and other knowledge points. The key is to find out the length of DE=BD and BD.
(20 13? As shown in the figure, it is known that △ABC is an equilateral triangle, points B, C, D and E are on the same straight line, CG=CD, DF=DE, then? E= 15 degrees.
Test center: the nature of equilateral triangle; Exterior angle property of triangle; Properties of isosceles triangle.
Analysis: According to the fact that the three angles of an equilateral triangle are equal, can you know? ACB=60？ That shows whether the base angles of isosceles triangles are equal. The degree of e.
Solution: Solution: ∫△ABC is an equilateral triangle.
ACB=60？ ,? ACD= 120？ ,
CG = CD，
CDG=30？ ,? FDE= 150？ ,
DF = DE，
E= 15？ .
So the answer is: 15.
Comments: This question examines the nature of an equilateral triangle. The sum of the two complementary angles is 180? The nature of isosceles triangle is moderately difficult.
(Zhanjiang, Guangdong, 20 13) As shown in the figure, one side of all regular triangles is parallel to the axis and one vertex is on the axis. From inside to outside, their side lengths are, and their vertices are represented by, where one unit is off-axis, and the coordinates of the vertices are.
Analysis: Investigate the knowledge of regular triangles and the ability to discover laws. As can be seen from the figure, the ordinate of is:
The abscissa of is: and the ordinate is, according to the meaning of the question. It is easy to find that,,, and these points are in the fourth quadrant, and the abscissa and ordinate are opposite. The subscripts of,,,, and are 2, 5, 7, 92, regular:, which is the right end point of the 3rd1th regular triangle (from inside to outside).
(20 13 Fuzhou, Fujian 19) As shown in the figure, in the plane rectangular coordinate system xOy, the coordinate of point A is (-2,0), and the equilateral triangle AOC can be translated, axisymmetric or rotated to get △OBD.
(1)△AOC translates to the right along the X axis to get △OBD, and the translation distance is one unit length; △AOC and△△ BOD are symmetrical about a straight line, then the symmetry axis is; △AOC rotates clockwise around the origin o to get△△ DOB, then the rotation angle can be degrees;
(2) Connect AD, cross OC at point E, and find? Degree of AEO
Test center: the nature of rotation; Properties of equilateral triangle; Axisymmetric property; The essence of translation.
Special topic: calculation problems.
Analysis: (1) If the coordinate of point A is (-2,0), according to the nature of translation, it is concluded that △AOC moves two units to the right along the X axis to get △OBD, then △AOC and △BOD are symmetric about Y; According to the properties of equilateral triangles? AOC=？ BOD=60？ And then what? AOD= 120？ According to the definition of rotation, △AOC rotates clockwise around the origin o 120? get△DOB；
(2) According to the nature of rotation, we get OA=OD, and? AOC=？ BOD=60？ , get? DOC=60？ Therefore, OE is the bisector of the vertex angle of isosceles triangle △AOD. According to the nature of isosceles triangle, OE bisects AD vertically, then? AEO=90？ .
Solution: Solution: (1) ∫ The coordinate of point A is (2,0).
? △AOC translates 2 units to the right along the X axis to get△△ OBD;
? △AOC and△△ BOD are symmetric about y;
∫△AOC is an equilateral triangle,
AOC=？ BOD=60？ ,
AOD= 120？ ,
? △AOC rotates clockwise around the origin o 120? Get delta delta △DOB.
(2) As shown in the figure, ∵ equilateral△ △AOC rotates clockwise around the origin o 120? Get delta delta △DOB,
? OA=OD，
∵? AOC=？ BOD=60？ ,
DOC=60？ ,
That is, OE is the bisector of the vertex angle of isosceles triangle △AOD,
? OE divides AD vertically,
AEO=90？ .
So the answer is 2; Y axis; 120.
Comments: This topic examines the essence of rotation: the congruence of two figures before and after rotation; The distance from the corresponding point to the rotation center is equal; The included angle between the corresponding point and the connecting line of the rotation center is equal to the rotation angle. The equilateral triangle, axial symmetry and translation are also studied.