Seven-grade mathematics knowledge points
Axisymmetry in life
1. Axisymmetric graph: If a graph is folded along a straight line and the parts on both sides of the straight line can completely overlap, then this graph is called an axisymmetric graph, and this straight line is called an axis of symmetry.
2. Axisymmetric: For two figures, if they can overlap each other after being folded in half along a straight line, then the two figures are said to be axisymmetric, and this straight line is the axis of symmetry. It can be said that these two figures are symmetrical about a straight line.
3. The difference between an axisymmetric figure and an axisymmetric figure: an axisymmetric figure is a figure, and an axisymmetric figure is the relationship between two figures.
Connection: They are all graphs folded along a straight line and can overlap each other.
2. Two symmetrical figures must be congruent.
3. Two congruent figures are not necessarily symmetrical.
The symmetry axis is a straight line.
5, the nature of the angle bisector
1, the straight line where the bisector of the angle is located is the symmetry axis of the angle.
2. Nature: the distance from the point on the bisector of the angle is equal to both sides of the angle.
6. perpendicular bisector of line segment
1, a straight line perpendicular to a line segment and bisecting the line segment is called the midline of the line segment, also called the midline of the line segment.
2. Property: the distance between the point on the vertical line in the line segment and the two ends of the line segment is equal.
7, axisymmetric graphics are:
Isosceles triangle (1 or 3), isosceles trapezoid (1), rectangle (2), diamond (2), square (4), circle (countless), line segment (1), angle (1), etc.
8, the nature of isosceles triangle:
① The two bottom angles are equal. ② The two sides are equal. 3 "three lines in one". (4) The height on the bottom edge and the line where the bisector of the center line and the vertex is located are its symmetry axis.
9.① Equiangular equilateral ∵∠B=∠C∴AB=AC.
② "equilateral angle" ∵ AB = AC ∴∠ B = ∠ C.
10, angle bisector property:
The point on the bisector of an angle is equal to the distance on both sides of the angle.
∫OA divides equally ∠CADOE⊥AC,OF⊥AD∴OE=OF.
1 1, the nature of the middle vertical line: the distance from the point on the middle vertical line to both ends of the line segment is equal.
∫oc vertically bisects AB∴AC=BC
12, the properties of axial symmetry
1. After two figures are folded in half along a straight line, the points that can overlap are called corresponding points, the line segments that can overlap are called corresponding line segments, and the angles that can overlap are called corresponding angles. 2. Two figures symmetrical about a straight line are congruent figures.
2. If two figures are symmetrical about a straight line, the line segments connected by corresponding points are vertically bisected by the symmetry axis.
3. If two figures are symmetrical about a straight line, then the corresponding line segment and the corresponding angle are equal.
13, mirror symmetry
1. When an object is placed in front of a mirror, the mirror will change its left and right direction;
2. When placed perpendicular to the mirror, the mirror will change its up-and-down direction;
3. If it is an axisymmetric figure, when the symmetry axis is parallel to the mirror, the image in the mirror is the same as the original figure;
Through discussion, students may find the following ways to solve the problem of mutual transformation between objects and images:
(1) Take photos with a mirror (pay attention to the placement of the mirror); (2) Using the axial symmetry property;
(3) Numbers can be reversed left and right, and simple axisymmetric figures can also be made;
(4) You can see the back of the image; (5) Imagine in your mind according to the previous conclusion.
Important knowledge points of seventh grade mathematics
Matters needing attention in specific methods of deformed names
The denominator is multiplied by the least common multiple of the denominator on both sides of the inequality (1). Items without denominator cannot be multiplied by ellipsis.
(2) Pay attention to the brackets of the fractional line. After removing the denominator, if the numerator is a polynomial, put parentheses.
(3) The number multiplied by both sides of inequality is negative, and the direction of inequality changes.
Depending on the meaning of the question, brackets can be removed from the inside out or from the outside in.
(1) When using the distribution law to remove brackets, don't omit the items in brackets.
(2) If there is a "-"before the brackets, the items in the brackets should be changed when the brackets are removed.
The shift term moves all the terms with unknowns to one side of the inequality (usually to the left), and the terms without unknowns to the other side of the inequality (crossing the bridge).
Merge similar items: merge similar items on both sides of the inequality respectively, and convert the inequality into the form of or.
Merging similar items only adds up the coefficients of similar items, and the letters and their indexes remain unchanged.
The coefficient of 1 is divided by the unknown coefficients on both sides of the inequality. If sum, the solution set of inequality is; If and, then the solution set of inequality is; If and, then the solution set of inequality is; If and, then the solution set of inequality is;
(1) The numerator and denominator cannot be reversed.
(2) Whether the inequality changes is determined by the positive and negative coefficients.
(3) Calculation sequence: calculate the values first, and then determine the symbols.
4. Representing the solution set of one-dimensional linear inequality on the number axis is an important embodiment of the idea of combining numbers with shapes in mathematics. Pay attention to the "three determinations": one is the demarcation point, the other is the direction, and the third is the emptiness.
5. The key to solving practical problems with one-dimensional linear inequality is to find the inequality relationship in the problem, so as to list inequalities and find the solution set of inequalities, and finally solve practical problems.
6. Basic linguistic meanings of common inequalities:
(1), then x is a positive number; (2), then x is negative;
(3), then x is not positive; (4), then x is nonnegative;
(5), then x is greater than y; (6), then x is less than y;
(7), then x is not less than y; (8), then x is not greater than y;
(9) Or, then X and Y have the same sign; (10) or, then x and y are different symbols;
(1 1)x and y are positive numbers, if, then; If so, then;
(12)x and y are both negative numbers, if, then; If, then
The first day of the second book mathematical triangle knowledge points
I. Objectives and requirements
1. Know the triangle, know the meaning of the triangle, know the sides, internal angles and vertices of the triangle, and express the triangle in symbolic language.
2. Experience the practical activities of measuring the side length of a triangle and understand the unequal relationship among the three sides of the triangle.
3. Know how to judge whether three line segments can form a triangle, and use it to solve related problems.
4. The interior angle theorem of triangle can be deduced from the properties of parallel lines.
5. Some simple practical problems can be solved by applying the triangle interior angle sum theorem.
Second, the main points
Theorem of sum of interior angles of triangle;
In order to understand the concept of triangle, three bars can be expressed in symbolic language.
Third, difficulties.
The reasoning process of triangle interior angle sum theorem;
Identify all triangles without repetition or omission in specific graphics;
Judging whether three line segments can form a triangle by the unequal relationship of three sides of a triangle.
Fourth, the knowledge framework.
Verb (abbreviation of verb) summary of knowledge points and concepts
1. triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.
2. Classification of triangles
3. Trilateral relationship of triangle: the sum of any two sides of triangle is greater than the third side, and the difference between any two sides is less than the third side.
4. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
5. midline: in a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the midline of the triangle.
6. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.
7. Significance and practice of high line, middle line and angle bisector.
8. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
9. Theorem of the sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.
It is inferred that the two acute angles of 1 right triangle are complementary;
Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles;
Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it;
The sum of the inner angles of a triangle is half of the sum of the outer angles.
10. External angle of triangle: the included angle between one side of triangle and the extension line of the other side is called the external angle of triangle.
1 1. The Properties of the Exterior Angle of Triangle
(1) Vertex is the vertex of a triangle, one side is one side of the triangle, and the other side is the extension line of one side of the triangle;
(2) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;
(3) The outer angle of a triangle is greater than any inner angle that is not adjacent to it;
(4) The sum of the external angles of the triangle is 360.
12. Polygon: On the plane, a figure composed of end-to-end line segments is called a polygon.
13. Interior angle of polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
14. Exterior angle of polygon: the angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
15. Diagonal line of polygon: the line segment connecting two non-adjacent vertices of polygon is called diagonal line of polygon.
16. Classification of polygons: it can be divided into convex polygons and concave polygons. Convex polygons can also be called plane polygons and concave polygons can also be called space polygons. Polygons can also be divided into regular polygons and non-regular polygons. Regular polygons have equal sides and equal internal angles.
17. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
18. plane mosaic: covering a part of a plane with some non-overlapping polygons is called covering the plane with polygons.
19. Formulas and attributes
The sum formula of polygon internal angles: the sum of n polygon internal angles is equal to (n-2) 180.
20. Polygon exterior angle sum theorem;
(1) The sum of the outer angles of n polygons is equal to n180-(n-2)180 = 360.
(2) Every inner angle of a polygon and its adjacent outer angle are adjacent complementary angles, so the sum of the inner angle and outer angle of n polygon is equal to n 180.
2 1. Number of diagonal lines of polygon:
(1) Starting from a vertex of an n polygon, (n-3) diagonal lines can be drawn, and the polygon can be divided into (n-2) triangles.
(2) An n-side * * has n(n-3)/2 diagonals.
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