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Authoritative PPT Summary
"Central Symmetry of Figures" PPT courseware
Knowledge link
rotation definition
In a plane, a figure is rotated by a certain angle in a certain direction (counterclockwise or clockwise) around a fixed point. This transformation is called the rotation of the figure. This fixed point is called the center of rotation, and this angle is called the angle of rotation.
properties of rotation
In a figure and the figure obtained by rotating it, the distances between the corresponding points and the center of rotation are equal; the angles formed by the two sets of corresponding points with the lines connecting the centers of rotation are equal.
Axisymmetric properties
The line connecting corresponding points in two axially symmetrical figures is bisected perpendicularly by the axis of symmetry
observe
(1) Rotate one of the patterns 180° around point O. What do you find?
(2) Line segments AC and BD intersect at point O, OA=OC, OB=OD. Rotate △OCD 180° around point O. What do you find?
Knowledge induction
Rotate a figure 180° around a certain point in a plane. This change in the figure is called central symmetry, and this fixed point is called the center of symmetry. If a figure can coincide with another figure through central symmetry, it is said that the two figures are about this fixed point. Be centrally symmetrical.
Central symmetry is a special case of rotation transformation. Two figures that are centrally symmetrical are congruent shapes.
Summary and summary
properties of central symmetry
(1) In two figures that are centrally symmetrical, the line connecting the corresponding points passes through the center of symmetry and is bisected by the center of symmetry.
(2) Two figures that are symmetric about the center are congruent figures.
Simple central symmetry graph
1. How to create a symmetrical point at the center of the point
Taking point O as the center of symmetry, draw the symmetry point A′ of point A;
Point A′ is the desired point
2. How to create a centrally symmetrical line segment
Taking point O as the center of symmetry, draw the symmetrical line segment point A′B′ of line segment AB.
case study
Example 1 As shown in the figure, △ABC and point O, draw △A′B′C′ that is symmetrical to △ABC about point O.
Analysis: How many points are needed to determine a triangle? To make a triangle that is centrally symmetrical about a certain point, how many points of symmetry do we need to make?
Drawing method: 1. Connect AO and extend it to A′, so that OA ′=OA, and get the symmetric point A′ of point A.
2. Similarly draw the symmetry points B′ and C′ of B and C.
3. Connect the points A′, B′, and C′ in sequence.
△A′B′C′ is the desired triangle.
How do you understand "the line segments connected to the symmetry point all pass through the symmetry center and are bisected by the symmetry center"?
Example 2 As shown in the figure, given the quadrilateral ABCD and point O, draw a figure that is centrally symmetrical with the quadrilateral ABCD about point O.
To draw a symmetrical figure of the quadrilateral ABCD with respect to point O, just draw the symmetric points A’.B’.C’.D’ of the four points A.B.C.D with respect to point O, and then connect the points in sequence.
Fill it out
1. The coordinates of the symmetry point of point P(1,3) about the x-axis are______
The coordinates of the symmetry point about the y-axis are_______
The coordinates of the symmetry point about the origin are _______.
2. It is known that point P(2a+b,a) and point P’(1,b) are symmetrical about the origin, then a=_____, b=_______.
Breakthrough in high school entrance examination
1. (Heze City High School Entrance Examination Question) It is known that points A (a-1, 5) and B (2, b-1) are symmetrical about the x-axis, then the value of (a+b)2006 is ( )
A. 0 B. -1
C. 1 D. (-3) 2006
2. (Shaanxi Provincial High School Entrance Examination Question) The coordinates of the symmetry point P1 of point P about the y-axis are (2, 3), then the coordinates of the symmetry point P2 of point P about the origin are ( )
A. (-3,-2) B. (2,-3)
C. (-2,-3) D. (-2,3)
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"Central Symmetry of Figures" PPT Courseware 2:
"Central Symmetry of Figures" PPT Courseware 2 Knowledge Link 1. Central Symmetry, Central Symmetry Rotate a figure 180 degrees around a certain point in the plane. This change in the figure is called central symmetry, and this fixed point is called the center of symmetry. A figure passes through Centrosymmetric..