Triangle:

Pythagorean Theorem: If the lengths of two right angles of a right triangle are A and B, and the length of the hypotenuse is C, then a2+b2=c2.

2. Inverse theorem of Pythagorean theorem: If the lengths of three sides are A, B and C, respectively, and a2+b2=c2, then this triangle is a right triangle.

3. A triangle means that the median line is parallel to the third side of the triangle and equal to half of the third side.

4. The midline of hypotenuse of right triangle is equal to half of hypotenuse.

Quadrilateral: Properties of (1) parallelogram

5. Two groups of opposite sides of a parallelogram are parallel (defined) respectively.

6. The opposite sides of the parallelogram are equal.

7. The diagonals of parallelograms are equal.

8. The diagonal bisection of parallelogram.

(2) Determination of parallelogram

9. Two groups of parallelograms with parallel opposite sides are parallelograms.

10, two groups of quadrilaterals with equal opposite sides are parallelograms.

Quadrilaterals whose diagonals bisect each other are parallelograms.

12, two sets of quadrilaterals with equal diagonals are parallelograms.

13. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

(C) the nature of the rectangle

14, all four corners of a rectangle are right angles.

15, the diagonals of the rectangles are equal.

(4) Determination of rectangle

16. A parallelogram with right angles is a rectangle. It is a rectangle. (definition)

A parallelogram with equal diagonal lines is a rectangle.

18. A quadrilateral with three right angles is a rectangle.

(5) the nature of diamonds

19, all four sides of the diamond are equal.

20. The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.

(vi) Determination of diamond shape

2 1, a set of parallelograms with equal adjacent sides is called a diamond. (definition)

22. Parallelograms with diagonal lines perpendicular to each other are diamonds.

23. A quadrilateral with four equilateral sides is a diamond.

(7) the nature of the square

24. All four sides are equal and all four corners are right angles.

25. Diagonal lines are vertically bisected and equal to each other.

(eight) the judgment of the square

26. A rectangle with equal adjacent sides is a square.

27. Diamonds with right angles are squares.

28. A quadrilateral with four equal sides and one angle at right angle is a square.

(9) Properties of isosceles trapezoid

29. The two angles on the same bottom of an isosceles trapezoid are equal.

30. The two diagonals of the isosceles trapezoid are equal.

Definition of (10) trapezoid

3 1, a set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.

32. An isosceles trapezoid is called an isosceles trapezoid. (can be judged)

33. A trapezoid with right angles is called a right trapezoid.

Determination of (1 1) isosceles trapezoid

34. A trapezoid with two equal angles on the same base is an isosceles trapezoid.

Center of gravity:

35. The center of gravity of the line segment is the midpoint of the line segment.

36. The center of gravity of a triangle is the intersection of three midlines.

37. The center of gravity of a parallelogram is the intersection of two diagonal lines.

38. The distance from the center of gravity to the vertex of a triangle is equal to twice the distance from the midpoint of the opposite side to the vertex.

quadratic radical

Generally speaking, we call the formal formula quadratic radical.

40. The properties of quadratic radical (1).

4 1, the properties of quadratic radical (2).

42, the nature of the quadratic radical (3).

43. Multiplication of quadratic roots. On the contrary, it can be used for quadratic radical simplification.

44. Division of quadratic roots. On the contrary, it can be used for quadratic radical simplification.

45. The simplest quadratic root: (1) takes the number of roots without denominator; (2) The number of roots does not include factors that can be turned on or off. The quadratic radical that satisfies the above two conditions is called the simplest quadratic radical.

46. Secondary roots with the same number of roots are called similar secondary roots.

47. When adding and subtracting secondary roots, you can first convert the secondary roots into the simplest secondary roots, and then merge the secondary roots with the same number of roots.

monadic quadratic equation

49. An equation containing an unknown number with the highest degree of 2 is called a quadratic equation.

50, the general form of quadratic equation:

5 1, an equation that can be solved by direct Kaiping method:

52. The matching method for solving a quadratic equation with one variable: (1) The constant term is shifted to the right (2) The two sides are divided by the quadratic term coefficient (3) The two sides are added with half of the square of the linear term coefficient.

53, the root formula of a quadratic equation:

54, the discriminant of quadratic equation root:

When δ > 0, the equation has two unequal real roots.

When δ = 0, the equation has two equal real roots.

When δ

55. Relationship between roots and coefficients of a quadratic equation (Vieta theorem): If two roots of a quadratic equation are x 1 and x2, then,

55, with x 1 and x2 as the root of the quadratic equation is

56. The basic idea of solving a quadratic equation with one variable is to simplify the quadratic equation with one variable into a quadratic equation with one variable. For some unary quadratic equations whose right side is 0 and whose left side can be decomposed into factors, if the left side can be decomposed into factors, these two factors can be used to reduce the order to solve the equations.

57, application type:

(1) The type of sum product: the sum and product of two numbers are known. To solve these two numbers, we can use (2) the problem of change rate: the formula is (3) the problem of area and invariance: the sum of the areas of each part is equal to the total area, such as the problem of repairing the path and framing the tablecloth (4) the problem of volume: the rectangular iron sheet is made by removing four corners.