Chapter I Evidence (2)
The "three lines in one" of isosceles triangle: the bisector of the top angle, the midline of the bottom edge and the height of the bottom edge coincide. ※.
An equilateral triangle is a special isosceles triangle, which is composed of three lines of an equilateral triangle and two congruent parts. ※
Right triangle, in which an acute angle is equal to 30? The right-angled side it faces must be equal to half of the hypotenuse.
Is there an angle equal to 60 degrees? ※? An isosceles triangle is an equilateral triangle.
If you know that a triangle is a right triangle, the first theorem to consider is:
① Pythagorean Theorem: (Pay attention to distinguish hypotenuse from right angle)
(2) In a right triangle, if there is an inner angle equal to 30? Then the right-angled side it faces is equal to half of the hypotenuse.
③ In a right triangle, the median line on the hypotenuse is equal to half of the hypotenuse (this theorem will appear in Chapter 3).
Perpendicular bisector is a straight line perpendicular to a line segment and bisecting the line segment. . (pay attention to the meaning of the bullet) ※
The point on the vertical line of a line segment is equal to the distance between the two endpoints of the line segment. ※.
Inverse theorem of the midline of a line segment: the point with equal distance to both ends of a line segment is on the midline of this line segment. ※.
The perpendicular bisector of three sides of a triangle intersect at a point, and the distance from the point to the three vertices is equal. . (as shown in figure 1, AO=BO=CO). ※
The distance between the point on the bisector of an angle and both sides of the angle is equal. ※.
Inverse theorem of angle bisector: If the distance between a point and both sides of the angle is equal, the point is on the bisector of the angle. ※.
The bisector of an angle is the set of all points with equal distance to both sides of the angle.
The three bisectors of a triangle intersect at a point, and the distances from the intersection point to the three sides are equal. The intersection point is the center of the triangle. ※.
(As shown in Figure 2, OD=OE=OF)
Chapter II Quadratic Equation in One Variable
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.
Solution of quadratic equation in one variable: ① collocation method. ※
(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) Two sides plus the square of half of the first coefficient;
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 Proof (III)
Definition of four parallel sides: A quadrilateral with two opposite sides parallel to each other is called a parallelogram, and a line segment connected by two vertices of a parallelogram is called its diagonal. ※.
The nature of parallelogram: the opposite sides of parallelogram are equal, the diagonal is equal, and the diagonal is equally divided. ※.
Distinguishing method of parallelogram: Two groups of parallelograms with parallel opposite sides are parallelograms. ※.
Two sets of quadrilaterals with equal opposite sides are parallelograms.
A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
A quadrilateral with two diagonal lines bisecting each other is a parallelogram.
Distance between parallel lines: If two lines are parallel to each other, the distance between any two points on one line and the other line is equal. This distance is called the distance between parallel lines. ※.
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.
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. ※.
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 IV Views and Forecast
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 closed wireframes connected 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 V 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 See if the product of two variables is a fixed value. (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.
Chapter VI Frequency and 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. ※.