Axisymmetric Mind Map of Mathematics in Senior Two Summarize the concept of knowledge 1. Basic concepts:
(1) Axisymmetric graph: If a graph is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this graph is called an axisymmetric graph.
⑵ Two graphs are symmetrical: one graph is folded along a straight line, and if it can overlap with another graph, the two graphs are said to be symmetrical about this straight line.
⑶ Midline of the line segment: The line passing through the midpoint of the line segment and perpendicular to the line segment is called the midline of the line segment.
⑷ isosceles triangle: A triangle with two equal sides is called an isosceles triangle. Two equal sides are called waist and the other side is called bottom. The angle between the two waists is called the top angle, and the angle between the buttocks and the waist is called the bottom angle.
5. equilateral triangle: A triangle with three equilateral sides is called an equilateral triangle.
2. Basic nature:
The essence of (1) symmetry;
(1) Whether an axisymmetric figure or two figures are symmetrical about a straight line, the axis of symmetry is arbitrary.
Perpendicular bisector of the line segment connected to the corresponding point.
② Symmetric figures are congruent.
(2) The nature of the vertical line in the line segment:
① The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.
② The point with the same distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(3) Coordinate properties of axisymmetrical points.
(4) the nature of isosceles triangle:
The waist of an isosceles triangle is equal.
② The two base angles of an isosceles triangle are equal (equilateral and equiangular).
③ The bisector of the top corner of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide.
④ The isosceles triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (1).
5] the properties of equilateral triangle:
All three sides of an equilateral triangle are equal.
② All three internal angles of an equilateral triangle are equal and equal to 60?
③ There are three lines on each side of an equilateral triangle.
An equilateral triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (3 lines).
3. Basic judgment:
Determination of (1) isosceles triangle;
A triangle with equal sides is an isosceles triangle.
(2) If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equiangular pair).
Equilateral).
(2) Determination of equilateral triangle:
A triangle with three equilateral sides is an equilateral triangle.
A triangle with three equal angles is an equilateral triangle.
③ There is an angle of 60? An isosceles triangle is an equilateral triangle.
4. Basic methods:
(1) perpendicular to a known straight line:
(2) The midline of the known line segment:
(3) Symmetry axis: connect two corresponding points and make the middle perpendicular of the connecting line segment.
(4) Make a symmetrical figure of a known figure about a straight line:
5. Make a point on the straight line so that the sum of the distances from it to two known points on the same side of the straight line is the shortest.
Book 11 of the old textbook of People's Education Press points out the definition of axial symmetry? If a graph is folded in half along a straight line, the graphs on both sides can completely overlap, and this graph is an axisymmetric graph. Su Jiaoban pointed out: If a figure is folded in half along a straight line, and the parts on both sides of the crease completely overlap after folding, then such a figure is called an axisymmetric figure. The picture of the comb is also an axisymmetric figure. Note: As long as the oblique figure can be folded along a straight line and the figures on both sides of the straight line can overlap each other, it is an axisymmetric figure. Draw a line in the middle of an axisymmetric figure, which is called the axis of symmetry.