A fifth grade math class on this chapter is always necessary for the student’s progress.
This lesson involves the following concepts:
– The definition
– property of conservation of lengths;
– conservation property of angle measures;
– conservation of the alignment;
– preservation of parallelism;
– transformation of a line into another parallel line;
– conservation of the perimeter of a figure;
– conservation of the area of a geometric figure.
The student should know how to construct the image of a figure by a central symmetry of center O but also how to use the various conservation properties to carry out demonstrations in geometry for the fifth level and throughout his schooling.
I. Definition of central symmetry
1. central symmetry and half-turn
Two figures and are symmetrical with respect to a point O when one can pass from one to the other by a half-turn of center O, that is to say a rotation of an angle of 180° and center O.
The figure is called the image of by the central symmetry of center O.
- The figure is the symmetrical of the figure with respect to the point O.
- Similarly, the figure is the symmetric of the figure with respect to the point O.
- The figures and are symmetrical with respect to the point O.
- We also say that the point O is the center of the symmetry that transforms the figure into the figure .
2.symmetric of a point
Consider a central symmetry of center O.
Point M’ is the image of point M by the central symmetry of center O
if and only if the point O is the middle of the segment [MM’].
- The symmetric of A with respect to O is A’.
- The symmetric of A’ with respect to O is A.
- A and A’ are symmetrical with respect to O.
II. properties of central symmetry
1.symmetric of a segment
- The symmetric of a segment by a central symmetry is a segment of the same length.
- Central symmetry preserves the lengths of segments, perimeters and areas of geometric figures.
- To construct the symmetry of the segment [CD] with respect to the point O, we construct the symmetry of the points C and D, noted C’ and D’, with respect to the point O.
- By the symmetry of center O, the symmetrical of the segment [CD] is then the segment [C’D’].
- The symmetric of the middle of a segment is the middle of the symmetric segment.
2.symmetry of a line
- The image of a line by a central symmetry is a line that is parallel to it.
- Central symmetry transforms a line into another line that is parallel to it.
3.symmetric of a polygon
Central symmetry preserves everything, mainly :
- the lengths;
- the perimeters of figures;
- areas of figures;
- angle measurements;
The symmetric of a polygon is a polygon with the same number of sides and the same shape.
To construct the symmetric of a polygon, we construct the symmetric of each side then,
we connect the vertices in the right order.
4.symmetrical of a circle
The symmetric of a circle is a circle with the same radius and having for center the symmetric of the center of the first circle.
To construct the symmetric of a circular arc with respect to a point, we construct the symmetries of the center and the ends of the symmetric circular arc.
III.center of symmetry of a figure
A point O is the center of symmetry of a figure if the image of the figure is coincident with .
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