High school compulsory mathematics knowledge points 1
Linear sum equation
(1) inclination angle of straight line
Definition: The angle formed by the positive direction of the X axis and the upward direction of the straight line is called the inclination angle of the straight line. Especially when the straight line is parallel or coincident with the X axis, we specify that its inclination angle is 0 degrees. Therefore, the range of inclination angle is 0 ≤ α.
(2) the slope of the straight line
① Definition: The tangent of a straight line whose inclination angle is not 90 is called the slope of the straight line. The slope of a straight line is often expressed by k, that is, the slope reflects the inclination of the straight line and the axis.
At that time,; At that time,; It didn't exist then.
② Slope formula of straight line passing through two points:
Pay attention to the following four points: (1) At that time, the right side of the formula was meaningless, the slope of the straight line did not exist, and the inclination angle was 90;
(2)k has nothing to do with the order of P 1 and P2; (3) The slope can be obtained directly from the coordinates of two points on a straight line without inclination angle;
(4) The inclination angle of a straight line can be obtained by calculating the slope through the coordinates of two points on the straight line.
(3) Linear equation
① Point-oblique type: the slope of the straight line is k, passing through the point.
Note: When the slope of the straight line is 0, k=0, and the equation of the straight line is y=y 1.
When the slope of the straight line is 90, the slope of the straight line does not exist, and its equation can not be expressed by point inclination. However, because the abscissa of each point on L is equal to x 1, its equation is x=x 1.
② Oblique section: the slope of the straight line is k, and the intercept of the straight line on the Y axis is b..
③ Two-point formula: () Two points on a straight line,
(4) Cutting torque type:
Where the straight line intersects the axis at the point and intersects the axis at the point, that is, the intercepts with the axis and the axis are respectively.
⑤ General formula: (A, B are not all 0)
Note: Various equations with special application scope, such as:
(4) A straight line parallel to the X axis: (b is a constant); A straight line parallel to the Y axis: (A is a constant);
(5) Linear system equation: that is, a straight line with some * * * property.
(1) parallel linear system
A linear system parallel to a known straight line (a constant that is not all zero): (c is a constant)
(2) Vertical linear system
A linear system perpendicular to a known straight line (a constant that is not all zero): (c is a constant)
(3) A linear system passing through a fixed point
(i) Linear system with slope k: a straight line passes through a fixed point;
(ii) The equation of the line system at the intersection of two lines is
(is a parameter), where the straight line is not in the straight line system.
(6) Two straight lines are parallel and vertical.
Note: When judging the parallelism and verticality of a straight line with slope, we should pay attention to the existence of slope.
(7) The intersection of two straight lines
stride
The coordinates of the intersection point are the solutions of a set of equations.
These equations have no solution; The equation has many solutions and coincidences.
(8) Distance formula between two points: Let it be two points in the plane rectangular coordinate system.
(9) Distance formula from point to straight line: distance from point to straight line.
(10) Distance formula of two parallel straight lines
Take any point on any straight line and then convert it into the distance from the point to the straight line to solve it.
Senior high school compulsory two mathematics knowledge points II
Structural characteristics of 1, column, cone, platform and ball
(1) prism:
Geometric features: the two bottom surfaces are congruent polygons with parallel corresponding sides; The lateral surface and diagonal surface are parallelograms; The sides are parallel and equal; The section parallel to the bottom surface is a polygon that is congruent with the bottom surface.
② Pyramid
Geometric features: the side and diagonal faces are triangles; The section parallel to the bottom surface is similar to the bottom surface, and its similarity ratio is equal to the square of the ratio of the distance from the vertex to the section to the height.
(3) Prism:
Geometric features: ① The upper and lower bottom surfaces are similar parallel polygons; ② The side is trapezoidal; ③ The sides intersect with the vertices of the original pyramid.
(4) Cylinder: Definition: It is formed by taking a straight line on one side of a rectangle as the axis and rotating the other three sides.
Geometric features: ① The bottom is an congruent circle; ② The bus is parallel to the shaft; ③ The axis is perpendicular to the radius of the bottom circle; ④ The side development diagram is a rectangle.
(5) Cone: Definition: A Zhou Suocheng is rotated with a right-angled side of a right-angled triangle as the rotation axis.
Geometric features: ① the bottom is round; (2) The generatrix intersects with the apex of the cone; ③ The side spread diagram is a fan.
(6) frustum of a cone: Definition: Take the vertical line of the right-angled trapezoid and the waist of the bottom as the rotation axis, and use Zhou Suocheng to rotate.
Geometric features: ① The upper and lower bottom surfaces are two circles; (2) The side generatrix intersects with the vertex of the original cone; ③ The side development diagram is an arch.
(7) Sphere: Definition: Geometry formed by taking the straight line with the diameter of the semicircle as the rotation axis and the semicircle surface rotating once.
Geometric features: ① the cross section of the ball is round; ② The distance from any point on the sphere to the center of the sphere is equal to the radius.
2. Three views of space geometry
Define three views: front view (light is projected from the front of the geometry to the back); Side view (from left to right),
Top view (from top to bottom)
Note: the front view reflects the height and length of the object; The top view reflects the length and width of the object; The side view reflects the height and width of the object.
3. Intuition of space geometry-oblique two-dimensional drawing method.
The characteristics of oblique bisection method are as follows: ① The line segment originally parallel to the X axis is still parallel to X, and its length remains unchanged;
② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.
4. Surface area and volume of cylinders, cones and platforms.
The surface area of a (1) geometry is the sum of all the surfaces of the geometry.
(2) The surface area formula of special geometry (C is the perimeter of the bottom, H is the height, and L is the generatrix)
(3) Volume formulas of cylinders, cones and platforms.
High school compulsory two mathematics knowledge points 3
Equation of circle
1. Definition of a circle: The set of points whose distance to a point on a plane is equal to a fixed length is called a circle, the fixed point is the center of the circle, and the fixed length is the radius of the circle.
2. Equation of circle
(1) standard equation, center and radius r;
(2) General equation
At that time, the equation represented a circle. At this point, the center is and the radius is.
At that time, I said a point; At that time, the equation did not represent any graph.
(3) Method of solving cyclic equation:
Generally, the undetermined coefficient method is adopted: first determine, then solve. Determining a circle requires three independent conditions. If the standard equation of a circle is used,
Demand a, b, r; If you use general equations, you need to find d, e, F e, f;
In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of a chord must pass through the origin, so as to determine the position of the center of the circle.
3. Summary of two compulsory knowledge points in high school mathematics: the positional relationship between straight line and circle:
The positional relationship between a straight line and a circle includes three situations: separation, tangency and intersection:
(1) Set a straight line and a circle, and the distance from the center of the circle to L is, then there is; ;
(2) Tangent to a point outside the circle: ①k does not exist, so verify the existence of ②k, establish an oblique equation, and solve k with the distance from the center of the circle to the straight line = radius, and get two solutions of the equation.
(3) The tangent equation of a point passing through a circle: circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through that point is (x0-a) (x-a)+(y0-b) (y-b) =
4. The positional relationship between circles: it is determined by comparing the sum (difference) of the radii of two circles with the distance (d) between the center of the circle.
Set a circle,
The positional relationship between two circles is usually determined by comparing the sum (difference) of the radii of the two circles with the distance (d) between the center of the circle.
At that time, the two circles were separated, and there were four common tangents at this time;
At that time, the two circles were circumscribed, and the connection line crossed the tangent point, with two outer tangents and one inner common tangent;
At that time, the two circles intersect, and the connecting line bisects the common chord vertically, and there are two external tangents;
At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent;
At that time, two circles included; It was concentric circles.
Note: when two points on the circle are known, the center of the circle must be on the vertical line in the middle; It is known that two circles are tangent and two centers are tangent to the tangent point.
5, spatial point, straight line, plane position relationship
Axiom 1: If two points of a straight line are on a plane, then all points of the straight line are on this plane.
Application: judging whether a straight line is in a plane.
Express axiom1in symbolic language;
Axiom 2: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.
Symbol: Plane α and β intersect, the intersection line is A, and it is denoted as α ∩ β = A. 。
Symbolic language:
The role of axiom 2:
It is a method to judge the intersection of two planes.
② Explain the relationship between the intersection line of two planes and the common point of two planes: the intersection line must pass through the common point.
③ It can be judged that a point is on a straight line, which is an important basis for proving several points.
Axiom 3: One and only one plane passes through three points that are not on the same straight line.
Inference: a straight line and a point outside the straight line determine a plane; Two intersecting straight lines define a plane; Two parallel lines define a plane.
Axiom 3 and its corollary: ① It is the basis for determining planes in space ② It is the basis for proving plane coincidence.
Axiom 4: Two lines parallel to the same line are parallel to each other.
High school compulsory two mathematics knowledge points 4
one
Structural characteristics of 1, column, cone, platform and ball
(1) prism:
Definition: Geometry surrounded by two parallel faces, the other faces are quadrangles, and the common edges of every two adjacent quadrangles are parallel to each other.
Classification: According to the number of sides of the bottom polygon, it can be divided into three prisms, four prisms and five prisms.
Representation: Use the letter of each vertex, such as a five-pointed star, or use the letter at the opposite end, such as a five-pointed star.
Geometric features: the two bottom surfaces are congruent polygons with parallel corresponding sides; The lateral surface and diagonal surface are parallelograms; The sides are parallel and equal; The section parallel to the bottom surface is a polygon that is congruent with the bottom surface.
② Pyramid
Definition: One face is a polygon, and the other faces are triangles with a common vertex. These faces enclose a geometric figure.
Classification: According to the number of sides of the bottom polygon, it can be divided into three pyramids, four pyramids and five pyramids.
Representation: Use the letters of each vertex, such as a pentagonal pyramid.
Geometric features: the side and diagonal faces are triangles; The section parallel to the bottom surface is similar to the bottom surface, and its similarity ratio is equal to the square of the ratio of the distance from the vertex to the section to the height.
(3) Prism:
Definition: Cut off the part between the pyramid, the section and the bottom with a plane parallel to the bottom of the pyramid.
Classification: According to the number of sides of the bottom polygon, it can be divided into triangular, quadrangular and pentagonal shapes.
Representation: Use the letters of each vertex, such as a pentagonal pyramid.
Geometric features: ① The upper and lower bottom surfaces are similar parallel polygons; ② The side is trapezoidal; ③ The sides intersect with the vertices of the original pyramid.
(4) Cylinder:
Definition: Geometry surrounded by a surface with one side of a rectangle and the other three sides rotating around a straight line.
Geometric features: ① The bottom is an congruent circle; ② The bus is parallel to the shaft; ③ The axis is perpendicular to the radius of the bottom circle; ④ The side development diagram is a rectangle.
(5) Cone:
Definition: Rotate the geometry surrounded by the surface of Zhou Suocheng with the right-angled side of the right-angled triangle as the rotation axis.
Geometric features: ① the bottom is round; (2) The generatrix intersects with the apex of the cone; ③ The side spread diagram is a fan.
(6) frustum of a cone:
Definition: Cut the part between the cone, the section and the bottom with a plane parallel to the bottom of the cone.
Geometric features: ① The upper and lower bottom surfaces are two circles; (2) The side generatrix intersects with the vertex of the original cone; (3) The side development diagram is an arch.
(7) Sphere:
Definition: Geometry formed by taking the straight line where the diameter of the semicircle is located as the rotation axis and the semicircle surface rotates once.
Geometric features: ① the cross section of the ball is round; ② The distance from any point on the sphere to the center of the sphere is equal to the radius.
2. Three views of space geometry
Define three views: front view (light is projected from the front of the geometry to the back); Side view (from left to right) and top view (from top to bottom)
Note: the front view reflects the position relationship of the object, that is, it reflects the height and length of the object;
The top view reflects the position relationship between the left and right of the object, that is, the length and width of the object;
The side view reflects the up-and-down and front-and-back positional relationship of the object, that is, it reflects the height and width of the object.
3. Intuition of space geometry-oblique two-dimensional drawing method.
The characteristics of oblique bisection method are as follows: ① The line segment originally parallel to the X axis is still parallel to X, and its length remains unchanged; ② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.
two
The positional relationship between two planes:
(1) The definition that two planes are parallel to each other: there is no common point between two planes in space.
(2) the positional relationship between two planes:
The two planes are parallel-have nothing in common; Two planes intersect-there is a straight line.
First, parallel
Theorem for determining the parallelism of two planes: If two intersecting lines in one plane are parallel to the other plane, then the two planes are parallel.
Parallel theorem of two planes: if two parallel planes intersect with the third plane at the same time, the intersection lines are parallel.
B, crossroads
dihedral angle
(1) Half-plane: A straight line in a plane divides this plane into two parts, and each part is called a half-plane.
(2) dihedral angle: The figure composed of two half planes starting from a straight line is called dihedral angle. The range of dihedral angle is [0, 180].
(3) The edge of dihedral angle: This straight line is called the edge of dihedral angle.
(4) Dihedral facet: These two half planes are called dihedral facets.
(5) Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the endpoint, and make two rays perpendicular to the edge in two planes respectively. The angle formed by these two rays is called the plane angle of dihedral angle.
(6) Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.
Esp。 The two planes are perpendicular.
Definition of two planes perpendicular: two planes intersect, and if the angle formed is a straight dihedral angle, the two planes are said to be perpendicular to each other. Write it down as X.
A theorem to determine the perpendicularity of two planes: If one plane passes through the perpendicular of the other plane, then the two planes are perpendicular to each other.
Verticality theorem of two planes: If two planes are perpendicular to each other, a straight line perpendicular to the intersection in one plane is perpendicular to the other plane.
three
pyramid
Definition of Pyramid: One face is a polygon and the other faces are triangles with a common vertex. The geometry surrounded by these faces is called a pyramid.
The essence of the pyramid:
The sides of (1) intersect at one point. The sides are triangular.
(2) The section parallel to the bottom surface is a polygon similar to the bottom surface. And its area ratio is equal to the square of the ratio of the height of the truncated pyramid to the height of the far pyramid.
Regular pyramid
Definition of a regular pyramid: If the bottom of the pyramid is a regular polygon and the projection of the vertex at the bottom is the center of the bottom, such a pyramid is called a regular pyramid.
The nature of the regular pyramid:
(1) An isosceles triangle whose sides intersect at one point and are equal. The height on the base of each isosceles triangle is equal, which is called the oblique height of a regular pyramid.
(3) Some special right-angled triangles
esp:
A For a regular triangular pyramid with two adjacent sides perpendicular to each other, the projection of the vertex on the bottom surface can be obtained as the vertical center of the triangle on the bottom surface by the three perpendicular theorems.
B there are three pairs of straight lines with different planes in the tetrahedron. If two pairs are perpendicular to each other, the third pair is perpendicular. And the projection of the vertex on the bottom surface is the vertical center of the triangle on the bottom surface.
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