I. Knowledge framework

Two. The concept of knowledge

1. rotation: in a plane, a graph rotates an angle around a graph in a certain direction, which is called graph rotation. This fixed point is called the center of rotation and the rotation angle is called the rotation angle. (The rotation of a graph means that every point on the graph moves around a fixed point on the plane by a fixed angle, in which the distance from the corresponding point to the rotation center is equal, the length of the corresponding line segment is equal to the size of the corresponding angle, and the size and shape of the graph have not changed before and after the rotation. )

2. Rotational symmetry center: A figure rotates an angle around a fixed point and then coincides with the original figure. This figure is called rotationally symmetric figure, this fixed point is called rotationally symmetric center, and the rotation angle is called rotation angle (rotation angle is less than 0 and greater than 360).

3. The central symmetry figure and central symmetry:

Centrally symmetric figure: If a figure can overlap itself after rotating 180 degrees around a certain point, then we say that this figure has formed a centrosymmetric figure.

Central symmetry: If one graph can overlap another graph after rotating 180 degrees around a certain point, then we say that these two graphs form central symmetry.

4. The essence of central symmetry:

On the congruence of two graphs with central symmetry.

For two graphs with central symmetry, the straight lines connecting the symmetrical points pass through and are equally divided by the symmetrical center.

For two figures with symmetrical centers, the corresponding line segments are parallel (or on the same straight line) and equal.

The content of this chapter is to let students understand the concept of rotation, explore the essence of rotation, further develop spatial observation, cultivate geometric thinking and aesthetic consciousness, experience the happiness of mathematics in practical problems and stimulate learning.

Chapter II: Circle

I. Knowledge framework

Two. The concept of knowledge

1. circle: A figure composed of all points whose distance from a plane to a fixed point is equal to a fixed length is called a circle. A fixed point is called the center of the circle and a fixed length is called the radius.

2. Arc chord: The part between any two points on the circle is called arc, or simply arc. An arc larger than a semicircle is called an upper arc, and an arc smaller than a semicircle is called a lower arc. Any one on the connected circle

A line segment representing two points is called a chord. The chord passing through the center of the circle is called the diameter.

3. Central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle.

4. Inner and outer center: the circle passing through the three vertices of the triangle is called the circumscribed circle of the triangle, and its center is called the outer center of the triangle. A circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, and its center is called the heart.

5. Sector: On a circle, the figure surrounded by two radii and an arc is called a sector.

6. The outline of the cone is a sector. The radius of this sector is called the generatrix of the cone.

7. The positional relationship between the circle and the point: Take the point P and the circle O as an example (let P be the point, then PO is the distance from the point to the center of the circle), and P is outside ⊙O, PO & gtr；; P on ⊙O，PO = r； P is within ⊙O, PO

8. There are three positional relationships between a straight line and a circle: there is no separated common point; There are two common points intersecting, and this straight line is called the secant of the circle; A circle and a straight line are tangent at a common point. This straight line is called the tangent of the circle, and this common point is called the tangent point.

9. There are five kinds of positional relations between two circles: if there is nothing in common, one circle is called outward separation from the other, and it is called inward separation; If there is a common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R, and R≥r, and the center distance is P: outward P>R+R; Circumscribed p = r+r; Intersecting R-r

10. Determination method of tangent: The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle.

1 1. The nature of the tangent: (1) The straight line perpendicular to this radius through the tangent point is the tangent of the circle. (2) The straight line perpendicular to the tangent point must pass through the center of the circle. (3) The tangent of the circle is perpendicular to the radius passing through the tangent point.

12. vertical diameter theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the two arcs opposite the chord.

13. related theorems:

The diameter (not the diameter) that bisects the chord is perpendicular to the chord and bisects the two arcs opposite the chord.

In the same circle or in the same circle, equal central angles have equal arcs and equal chords.

In the same circle or in the same circle, the central angle of the same arc and the same arc is equal, which is equal to half of the central angle of the arc.

The semicircle (or diameter) is opposite to the right angle, and the chord is opposite to the 90-degree angle.

14. Calculation formula of circle 1 Circumference C=2πr=πd2. Area s = π r 2; 3. Sector arc length l=nπr/ 180

15. Sector area s = π (r 2-r 2) 5. Transverse area of cone S=πrl.

Chapter 3: One-variable quadratic radical

I. Knowledge framework

Two. The concept of knowledge

One-dimensional quadratic equation: An equation with algebraic expression on both sides and only one unknown (one-dimensional) and unknown degree of 2 (quadratic) is called one-dimensional quadratic equation.

Generally speaking, any univariate quadratic equation about X can be transformed into the following form ax2+bx+c=0(a≠0) after sorting. This form is called the general form of quadratic equation with one variable.

The order of the unary quadratic equation is ax2+bx+c=0(a≠0), where ax2 is the quadratic term and A is the coefficient of the quadratic term; Bx is a linear term, and b is a linear term coefficient; C is a constant term.

The content of this chapter mainly requires students to solve some practical problems by solving equations on the premise of understanding quadratic equations in one variable.

(1) Solve the equation with the shape of (x+m)2=n(n≥0) by Kaiping method; Understand the mathematical thought of degeneration-transformation.

(2) The general steps of solving a quadratic equation with one variable by using the matching method: the known equation is now transformed into a general form; The coefficient of the quadratic term is1; Constant term moves to the right; Add half the square of the coefficient of the first term to both sides of the equation to make the left side completely equal; The deformation form is (x+p)2=q, if q≥0, the root of the equation is x =-p √ q; If you ask

When introducing the matching method, the shape equation is first deduced through practical problems. Such an equation can be transformed into a simpler equation, and the solution of this equation can be obtained from the concept of square root. Then an example is given to illustrate how to solve the equation in the form of. Then an example is given to show that the quadratic equation of one variable can be transformed into a formal equation, and the matching method is derived. Finally, an example of solving a quadratic equation with one variable by collocation method is given. The example involves a quadratic equation whose quadratic coefficient is not 1, and also involves a quadratic equation without real number roots. For the quadratic equation with one variable without real number roots, students will have a further understanding of this content after learning the formula method.

(3) The root of the unary quadratic equation ax2+bx+c=0(a≠0) depends on the coefficients A, B and C of the equation, so:

When solving a quadratic equation with one variable, we can first change the equation into the general form ax2+bx+c=0. When b2-4ac≥0, we can substitute A, B and C into the formula x= to get the root of the equation. (The operations in the formula just include the six operations of addition, subtraction, multiplication, division, multiplication and square root that we have learned, so it can be seen that this formula is called the formula for finding the root of a quadratic equation with one variable. The method of solving a quadratic equation with one variable by finding the root formula is called formula method.