I. Numbers and formulas

For example, the square root of is. (a) 2、(b)、(c)、(d)。

Example: The equation holds. (a)、(b)、(c)、(d)。

Second, the equation and inequality

(1) letter coefficient

Example: the equation about, and. Prove: Equations always have real roots.

Example: If the solution set of the inequality group is, then the value range of is.

(A)、(B)、(C)、(D)。

(2) Discrimination

Example: It is known that a quadratic equation with one variable has two real roots, which satisfies the realistic range of inequality and number.

⑶ Definition of solution

Example: If the real number is known and the conditions are met, then = _ _ _ _ _ _ _ _.

(4) increase the root system

Example: Why is there no real solution when the value is zero?

5] Application background

Example: someone goes down to the ground by boat, and then goes upstream to the ground. It takes three hours by boat. It is known that the speed of the ship in still water is 8 km/h, the current speed is 2 km/h and the distance between the two places is 2 km. Find the distance between them.

[6] Root deletion

Example: solving equations.

Third, function

(1) independent variable

Example: In the function, the value range of the independent variable is _ _ _ _ _ _ _ _ _.

⑵ letter coefficient

Example: If the image of the quadratic function passes through the origin, then = _ _ _ _ _ _ _ _ _ _.

⑶ Functional image

Example: If the range of the independent variable of a linear function is, and the range of the corresponding function value is, find the analytic function.

⑷ Application background

For example, a hotel has 65,438+000 beds. When the charge for each bed is 65,438+00 yuan per night, all rooms can be rented. If the charge per bed per night is increased by 2 yuan, 10 beds will be rented. If you change this method to improve 2 yuan every time, you will get benefits with less investment, and each bed will be improved by _ _ _ _.

Fourth, linear type.

(1) Unknown reference

Example: If the sum of two sides of a right triangle, the height on the hypotenuse is greater than _ _ _ _ _.

(2) similar triangles's corresponding problem.

Example: In,, and, it is the last point. Take a point on the table and get it. If two triangles are similar, find the length.

(3) The base of an isosceles triangle.

Example: If one side of an isosceles triangle is 4 and its perimeter is 10, its area is _ _ _ _ _.

(4) The height of triangle.

For example, if one side of an isosceles triangle is 10 and its area is 25, what is the vertex angle of the triangle?

5] Rectangular problem

Example: There is a triangular iron sheet with the longest side = 12cm and the height =8cm. It needs to be processed into a rectangular iron sheet, so that one side of the rectangle is on the top, and the other two vertices are on the other sides of the triangle, and the length of the rectangle is twice as wide. What is the area of the processed iron sheet?

[6] the problem of proportion

For example: If, then = _ _ _ _ _.

Error-prone problems in verb (verb abbreviation) circle

The positional relationship between (1) point and chord.

Example: The diameter of ⊙O is known, and the point is on ⊙ O. The vertical line leading to the diameter passes through this point, and the vertical foot is the point. If the radius ⊙O is equal to 5, then = _ _ _ _ _.

(2) the position relationship between point and arc.

Example:, is the tangent of ⊙O,, is the tangent point, and the point is any point different from, so _ _ _ _.

(3) The positional relationship between the parallel chord and the center of the circle.

Example: There are two parallel chords on a circle with a radius of 5cm, the lengths of which are 6cm and 8cm respectively, so the distance between these two chords is equal to _ _ _ _ _ _ _ _.

(4) the positional relationship between the intersecting chord and the center of the circle

Example: If the chord length of two intersecting circles is 6 and the radii of the two circles are 0 and 5 respectively, then the distance between the centers of the two circles is equal to _ _ _ _ _ _ _.

5] The positional relationship of tangent circle.

Example: If the radii of two concentric circles are 2 and 8 respectively, and the third circle is tangent to the two circles respectively, the radius of this circle is _ _ _ _ _ _.

Exercise questions:

First, the topics that are easy to miss.

1. If the absolute value of a number is 5, then this number is _ _ _ _ _ _ _; The absolute value of _ _ _ _ _ number is itself. (,non-negative)

2. The reciprocal of _ _ _ _ _ is itself; The cube of _ _ _ _ _ is itself. (,and 0)

3. Positive integer solution of inequality about 1 and 2; The value range of is _ _ _ _ _ _ _. ()

4. If the solution set of the inequality group is, then the value range of is _ _ _ _ _ _ _. ()

5. If yes, then _ _ _ _ _ _ _. (,2,, 0)

6. When it is a value, the function is a linear function. (or)

7. If all three sides of a triangle are solutions of the equation, then the perimeter of the triangle is _ _ _ _ _ _ _. (12, 24 or 20).

8. If the real number is satisfied, then _ _ _ _ _. (2,)

9. Draw four points on a plane at will, and then these four points * * * can determine _ _ _ _ straight lines.

10. If it is known that the line segment =7cm and the line segment drawn on the straight line =3cm, then the line segment = _ _ _. (4 cm or 10 cm).

1 1. Two sides of one angle and two sides of another angle are perpendicular to each other, and one angle is twice as large as the other. Find the degrees of these two angles. (,or,)

12. Three straight roads cross each other to form a triangle. Now, if we want to build a cargo transfer station, which requires the same distance from three highways, what are the alternative addresses? (4)

13. The ratio of waist height to waist length of an isosceles triangle means that the vertex angle of the triangle is _ _ _ _ _. (or)

14. The waist length of an isosceles triangle is, and the included angle between the height of one waist and the height of the other waist is, so the height on the bottom of the isosceles triangle is _ _ _ _ _ _. (or)

15. The diagonal of the rectangle intersects this point. If one side is a regular triangle with a length of 1, the perimeter of this rectangle is _ _ _ _ _. (or)

16. In the trapezoid,, =7cm, =3cm, try to determine the position of the side, so that the triangle with,, as the vertex is similar to the triangle with,, as the vertex. (= 1 cm, 6 cm or cm)

17. Given a line segment = 10cm, the distances from the endpoints to the straight lines are 6cm and 4cm respectively, and there are _ _ _ eligible straight lines. (3 lines)

18. There are _ _ _ _ * circles intersecting with two points outside the straight line whose center is on the straight line. (0, 1 or countless)

19. In,,,, there is only one intersection point between the circle with center and radius and the hypotenuse, and the range of values is obtained. (or)

20. In the rectangular coordinate system, how many points on the known axis make it an isosceles triangle? (4)

2 1. In the same circle, the relationship between the circumferential angles of chords is _ _ _ _ _ _ _ _. (equal or complementary).

22. The radius of the circle is 5cm, and the lengths of the two parallel chords are 8cm and 6cm respectively, so the distance between the two parallel chords is _ _ _ _ _ _ _ (1cm or 7cm).

23. The radii of two concentric circles are 9 and 5 respectively. If a circle is tangent to both circles, what is the radius of the circle? (2 or 7)

24. A circle is tangent to a circle with a radius of 5, and the distance between the centers of the two circles is 3. What is the radius of this circle? (2 or 8)

25. The tangent ⊙O is the chord of ⊙O, and if the radius of ⊙O is 1, the length of ⊙ o is _ _ _ _. (1 or).

26., is the tangent of ⊙O, is the tangent point, and the point is any point different from, so _ _ _ _ _. (or)

27. In ⊙O with a radius of 1, the chord is _ _ _ _ _ _. (or)

Second, the problem that is easy to solve.

28. If it is known, then _ _ _ _. (3)

29. In the function, the range of independent variables is _ _ _ _ _. ()

30. If known, then _ _ _ _ _. ()

3 1. When it is a value, the equation about has two real roots. (,and).

32. When the value is what, the function is a quadratic function. (2)

33. If, then? . ( )

34. What is the number of groups of real number solutions of the equation? (2)

35. The equation has a real number solution and the range of values is obtained. ()

36. Why is the sum of the squares of the two roots of the equation 23? ( )

37. Why are the two roots of the equation just the cosine values of the two acute angles of a right triangle? . ( ).

38. If the score is meaningful to any real number, the value of should satisfy _ _ _ _ _. ().

39. How many quadrilaterals can you make by doing,,, and in the middle? ( 1)

40. In ⊙O, the chord =8cm is a point on the chord, and =2cm. What is the shortest chord length passing through this point? (cm)

4 1. Two coins are always in contact, one is fixed and the other is rolling around it. When the rolling coin rolls around the fixed coin and returns to its original position, the number of turns of the rolling coin is _ _ _ _ _. (2)

Three, easy to misjudge the problem:

1. The heights of two sides and a set of opposite sides correspond to the coincidence of two triangles.

2. The heights of two sides and the third side correspond to the congruence of two triangles.

3. Two angles and two triangles with equal sides.

4. The diagonal of two sides and one side corresponds to the combination of two equal triangles.