1. 1 the nature and judgment of diamonds

Definition of rhombus: A group of parallelograms with equal adjacent sides is called rhombus.

The nature of the diamond: it has the nature of a parallelogram, four sides are equal, two diagonals are bisected vertically, and each diagonal bisects a set of diagonals. ※.

The diamond is an axisymmetric figure, and the straight line where each diagonal line is located is the axis of symmetry.

The distinguishing method of rhombus: A group of parallelograms with equal adjacent sides is rhombus. ※.

Parallelograms with diagonal lines perpendicular to each other are diamonds.

A quadrilateral with four equilateral sides is a diamond.

Properties and Judgment of 1.2 Rectangle

Definition of rectangle: A parallelogram with a right angle is called a rectangle. A rectangle is a special parallelogram. ※.

The nature of rectangle: it has the nature of parallelogram, with equal diagonals and four corners at right angles. A rectangle is an axisymmetric figure with two axes of symmetry. ※. )

Determination of rectangle: A parallelogram with a right angle is called a rectangle (by definition). ※.

A parallelogram with equal diagonal lines is a rectangle.

A quadrilateral with four equal angles is a rectangle.

Inference: The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse. ※.

The Property and Judgment of 1.3 Square

Definition of a square: A group of rectangles with equal adjacent sides is called a square.

Properties of Square: Square has all the properties of parallelogram, rectangle and diamond. A square is an axisymmetric figure with two axes of symmetry. ※.

Common judgment of a square: a diamond with a right angle is a square;

A rectangle with equal adjacent sides is a square;

The rhombus with equal diagonal lines is a square;

A rectangle with diagonal lines perpendicular to each other is a square.

The relationship between squares, rectangles, diamonds and parallel edges (as shown in Figure 3):

Definition of trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid. ※.

Two trapezoid with equal waist are called isosceles trapezoid. ※.

A trapezoid with a vertical waist bottom is called a right-angled trapezoid. ※.

The nature of isosceles trapezoid: the two internal angles on the same bottom of isosceles trapezoid are equal and the diagonal lines are equal. ※.

Two trapeziums with equal internal angles on the same base are isosceles trapeziums.

The center line of the triangle is parallel to the third side and equal to half of the third side. ※.

The parallel lines sandwiched between two parallel lines are equal. ※.

In a right triangle, the center line of the hypotenuse is equal to half of the hypotenuse. ※

Chapter II Quadratic Equation in One Variable

2. 1 Understanding the quadratic equation of one variable

2.2 Using collocation method to solve quadratic equation

2.3 Solving quadratic equation with formula method

2.4 factorization method to solve a quadratic equation.

2.5 the relationship between the quantity and the coefficient of a quadratic equation.

2.6 the application of quadratic equation

The whole equation contains only one unknown, and they can all be reduced to (A, B and C are. ※

Constant, a≠0), such an equation is called quadratic equation.

(a, b, c are constants, and a≠0) is called the general form of a quadratic equation, and a is the coefficient of the quadratic term. B is linear coefficient. ※: C is a constant term.

Methods to solve the quadratic equation of one variable: ① collocation method, that is, the form it becomes >

(2) Formula method (note that when looking for abc, you must first turn the equation into a general form)

③ Factorization method turns one side of the equation into 0 and the other side into the product of two linear factors to solve it. (mainly including "improving common factor" and "cross multiplication")

The basic steps of solving a quadratic equation with one variable by matching method are: ① transforming the equation into a general form of a quadratic equation with one variable; ※:

② Convert the quadratic coefficient into1;

③ Move the constant term to the right of the equation;

(4) Add the square of half of the first coefficient on both sides;

⑤ Transform the equation into the form of;

⑥ Find the root of both sides.

Relationship between roots and coefficients: When B2-4ac > ※: 0, the equation has two unequal real roots;

When b2-4ac=0, the equation has two equal real roots;

When B2-4ac

※ If the two roots of the unary quadratic equation are x 1 and x2, then:.

Function of the relationship between roots and coefficients of a quadratic equation with one variable. ※:

(1) Know one equation and find another;

(2) Solve the equation, and find the value of the symmetric formula of the roots of the quadratic equation x 1 and x2, paying special attention to the following formula:

① ② ③

④ ⑤

⑥ Other algebraic expressions that can be used or expressed.

(3) Given two x 1 and x2 of the equation, we can construct an unary quadratic equation:

(4) Given the sum and product of two numbers x 1 and x2, the problem of finding these two numbers can be transformed into finding the root of a quadratic equation.

When solving application problems with equations, there are two main steps: ① setting unknowns (when setting unknowns, in most cases, only the problem with the purpose of X is solved; But sometimes it must be considered according to known conditions, equivalence relations and other aspects); (2) Find equivalence relation (a general topic will contain a sentence expressing equivalence relation, and only need to find this sentence to list equations according to it).

The process of solving the problem can be further summarized as follows: ※:

Chapter III Further Understanding of Probability

3. 1 Use tree diagram or table to find probability.

3.2 Use frequency to estimate probability

In the frequency distribution table, the number of each group of data is called frequency. ※:

The ratio of each group of frequencies to the total data is called this group of frequencies; Namely:

In the frequency distribution histogram, the area of each small rectangle is equal to the frequency of each group, and the sum of each group's frequencies is equal to 1. Therefore, the sum of the areas of each small rectangle is equal to 1.

Frequency distribution table and frequency distribution histogram are two different representations of frequency distribution of a group of data, the former is accurate and the latter is intuitive. ※.

Use the frequency of an event to estimate the probability of this event.

Probability can be obtained by list method, but this method is not suitable for more complicated situations.

Assuming that there are m black balls in the bag, we can estimate the probability that a ball randomly found in the bag is a white ball through many experiments. ※:

To estimate the number of fish in the pond, you can first catch 100 fish from the pond as a mark, then put them back into the pond and catch 200 fish from the pond. If there are 10 fish marked, and then there are x fish in the pond, the number of fish can be estimated accordingly. . (Note that the estimated data is inaccurate, so it should be called "around XX". ※

There are a lot of uncertain events in life. Probability is a mathematical model to describe uncertain phenomena, which can accurately measure the possibility of events, but it does not mean that they will happen. ※.

Probabilistic solution:

(1) Generally speaking, if there are n possible results in an experiment, and their probability of occurrence is equal, and event A contains m results, then the probability of occurrence of event A is P(A)= 1

(2) List method

The method of analyzing and solving the probability of some events by list method is called list method.

(3) Tree diagram method

By listing all possible results of an event in a tree diagram, the method of finding its probability is called tree diagram method.

When an experiment is designed with three or more factors, it is inconvenient to use the list method. In order to list all possible results without weighting or omission, the tree diagram method is usually used to find the probability. )

Chapter IV Graphic Similarity

4. 1 proportional line segment

4.2 Parallel segments are proportional

4.3 Shaped like a polygon

4.4 Explore the conditions of triangle similarity

4.5 Proof of similar triangles's Judgment Theorem

4.6 Use similar triangles to measure height.

4.7 Nature of similar triangles

4.8 Graphic Similarity

1. the ratio of line segments.

1. If the same length unit is used to measure two line segments AB, and the length of CD is m and n respectively, then the ratio of these two line segments AB, CD = m: n can be said or written. ※ 。

2. Among the four line segments A, B, C and D, if the ratio of A to B is equal to the ratio of C to D, that is, these four line segments are called proportional line segments for short. ※ 。

※ 3. Note:

①a:b=k, which means that A is k times that of B;

(2) Because the lengths of line segments A and B are both positive numbers, k is a positive number;

③ The ratio has nothing to do with the length units of the selected line segments, and the length units of the two line segments should be consistent when solving;

④ Except a=b, a:b≠b:a is reciprocal;

⑤ Basic properties of proportion: If yes, then ad = bc If ad=bc, then

2. The golden section

1. As shown in figure 1, point C divides the line segment AB into two lines, AC and BC. If so, it is said that the line segment AB is divided by the golden section of point C, which is called the golden section of the line segment AB, and the ratio of AC to AB is called the golden section ratio. ※ 。

2. The golden section is the most beautiful and pleasing point. ※ 。

Four. similar polygons

1.Generally, graphics with the same shape are called similar graphics.

2. Two polygons with equal corresponding angles and proportional corresponding sides are called similar polygons. The ratio of corresponding edges of similar polygons is called similarity ratio. ※ 。

Verb (abbreviation of verb) similar triangles

1. Among the similar polygons, similar triangles is the simplest one. ※ 。

2. A triangle with equal corresponding angles and proportional corresponding sides is called similar triangles. The ratio of corresponding edges in similar triangles is called similarity ratio. ※ 。

3. congruent triangles is a special case of similar triangles, when the similarity ratio is equal to 1. Note: Just like two congruent triangles, the letters representing the corresponding vertices should be written in the corresponding positions. ※ 。

4. The similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio. ※ 。

5. The ratio of the circumference of similar triangles is equal to the similarity ratio. ※ 。

6. The ratio of similar triangles area is equal to the square of similarity ratio. ※ 。

6. Explore the conditions of triangle similarity

1. similar triangles's judgment method. ※:

Ordinary triangle right triangle

Fundamental Theorem: A straight line parallel to one side of a triangle and intersecting with the other two sides (or extension lines of both sides) is similar to the original triangle.

(1) The two angles are equal;

(2) The two sides are proportional and the included angle is equal;

③ Three sides are proportional. ① An acute angle is equal;

(2) Both sides are proportional:

A. two right-angled sides are proportional;

B. The hypotenuse is proportional to the right angle.

2. Proportional theorem of parallel lines divided into segments: three parallel lines cut two straight lines, and the corresponding segments are proportional. ※ 。

As shown in figure 2, l1/L2//L3, then.

3. The straight line parallel to one side of the triangle intersects with the other two sides (or the extension lines of both sides), and the triangle formed is similar to the original triangle. ※ 。

Eight. Properties of similar polygons

The perimeter of a similar polygon is equal to the similarity ratio. The area ratio be equal to the square of the similarity ratio. ※.

Nine. Magnification and reduction of graphics

1. If two graphs are not only similar graphs, but also the straight lines of each group of corresponding points pass through the same point, then such two graphs are called potential graphs. This point is called potential center. ※: At this time, the similarity ratio is also called similarity ratio.

2. The ratio of the distance between any pair of corresponding points and the center of the potential diagram is equal to the potential ratio. ※ 。

◎3. Potential changes:

① The transformed graph is not only similar to the original graph, but also the connecting lines of the corresponding vertices intersect at one point, and the distance between the corresponding points and this intersection point is proportional. This special similarity transformation is called potential transformation, and this intersection point is called potential center.

(2) A graph is transformed into another graph by potential, and these two graphs are called potential shapes.

(3) Using analogy method, we can enlarge or reduce the figure.

Chapter V Projections and Views

5. 1 projection

5.2 view

The three views include front view, top view and left view. ※.

Keep the three views aligned, high level and equal width. Generally, the top view should be drawn below the front view, and the left view should be drawn on the right side of the front view.

Front view: an image seen from the front of an object.

Top view: Basically, it can be considered as an image seen from above an object.

Left view: an image seen from the left side of an object.

※ Each closed wireframe in the view represents a face (plane or surface) on the object, and the two connected closed wireframes must not be on the same plane.

※ Each small wireframe contained in the outline box must be each small plane (or surface) protruding or recessed on a plane (or surface).

When drawing a view, the outline of the visible part is usually drawn as a solid line, and the outline of the invisible part is usually drawn as a dotted line. ※.

When an object is illuminated by light, it will leave its own shadow on the ground or on the wall. This is a projection.

Solar rays can be regarded as parallel rays, and the projection formed by such rays is called parallel projection.

The light of searchlights, flashlights and street lamps can be regarded as starting from a point, and the projection formed by this light is called central projection.

The difference between parallel projection and central projection: ① observing the light source; ② Observe the shadows.

The position of the eyes is called the viewpoint; The line from the viewpoint is called the line of sight; The place where the eyes can't see is called blind spot.

The figure seen from the front, top and side is a common orthographic projection, which is the projection when the light is perpendicular to the projection. ※.

(1) the projection of a point on the plane is still a point;

(2) the projection of the line segment on the plane can be divided into three situations:

When the line segment is perpendicular to the projection plane, the projection is a point;

When the line segment is parallel to the projection plane, the projection length is equal to the actual length of the line segment;

When the line segment is inclined to the projection plane, the projection length is less than the actual length of the line segment.

(3) The projection of the plane figure on the plane can be divided into three situations:

When the plane figure is parallel to the projection plane, its projection is the actual shape;

When the plane figure is perpendicular to the projection plane, its projection is a line segment;

When the plane figure and the projection plane are inclined, the projection is smaller than the actual shape.

Chapter VI Inverse Proportional Function

6. 1 inverse proportional function

6.2 Images and Properties of Inverse Proportional Function

6.3 Application of Inverse Proportional Function

The concept of inverse proportional function: Generally speaking, (k is a constant, k≠0) is called inverse proportional function, that is, y is the inverse proportional function of x. ※. (x is an independent variable and y is a dependent variable, where x cannot be zero)

Equivalent form of inverse proportional function: y is the inverse proportional function of x ←→→→→→→ variable y is inversely proportional to x, and the proportional coefficient is K. ※ 。

There are two ways to judge whether two variables are inverse proportional functions: ① According to the definition of inverse proportional function; (2) to see whether the product of two variables is a constant value, that is >. (Usually the second method is more suitable)

The image of inverse proportional function consists of two curves, which are called hyperbola. ※

Matters needing attention in drawing inverse proportional function: ① The image of inverse proportional function is not a straight line, so "two-point method" cannot be drawn;

② The more points selected, the more accurate the drawing;

③ Pay attention to its aesthetics (symmetry and ductility) when drawing.

Properties of inverse proportional function. ※:

(1) when k >; 0, the two branches of the hyperbola are located in the first and third quadrants respectively; In each quadrant, y decreases with the increase of x;

② when k

③ The two branches of hyperbola will be infinitely close to the coordinate axis (X axis and Y axis), but they will not intersect with the coordinate axis.

Geometric characteristics of inverse proportional function image: (as shown in Figure 4) ※

The point P(x, y) exists on the hyperbola.