(1) geometry

1. Structural characteristics of column, cone, platform and ball

(1) column

Prism: Generally, two faces are parallel to each other, and the other face is a quadrangle, and the public sides of every two adjacent quadrangles are parallel to each other. The geometry surrounded by these faces is called a prism; The two parallel faces in the prism are called the bottom face of the prism, which is called the bottom face for short; The other faces are called the sides of the prism; The common side of the adjacent side is called the side of the prism; The common vertex of the edge and the bottom is called the vertex of the prism.

Prisms with triangular, quadrilateral and pentagonal bottoms are called triangular prisms, quadrangular prisms and pentagonal prisms respectively.

Cylinder: The geometry surrounded by a curved surface formed by the rotation of a straight line on one side of a rectangle is called a cylinder; The axis of rotation is called the axis of the cylinder; The surface formed by rotating a surface perpendicular to the axis is called the side of the cylinder; No matter where it rotates, the surface that is not perpendicular to the axis is called the generatrix surface of the cylinder.

Prisms and cylinders are collectively referred to as cylinders;

(2) Cone

Pyramid: Generally, 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. This polygonal surface is called the bottom of the pyramid; Each triangular face with a common vertex is called the edge of the pyramid; The common vertex of each side is called the vertex of the pyramid; The common side of adjacent sides is called the side of a pyramid.

Prisms with triangular pyramid, quadrangular pyramid and pentagonal pyramid at the bottom are called triangular pyramid, quadrangular pyramid and pentagonal pyramid respectively.

Cone: The geometry surrounded by a surface formed by rotating one right-angled side of a right-angled triangle as the rotation axis and the other two sides is called a cone; The rotating shaft is a cone; The surface formed by the rotation of the surface perpendicular to the axis is called the bottom surface of the cone; The curved surface formed by the rotation of the hypotenuse is called the side of the cone.

Pyramids and cones are collectively called cones.

(3) Taiwan Province

Prism: cut the pyramid with a plane parallel to the bottom surface, and the part between the bottom surface and the cross section is called prism; The bottom surface and cross section of the original pyramid are called the lower bottom surface and the upper bottom surface of the pyramid respectively; Prisms also have sides, sides and vertices.

Cone: cut the cone with a plane parallel to the bottom, and the part between the bottom and the section is called cone; The bottom surface and cross section of the original cone are respectively called the lower bottom surface and the upper bottom surface of the frustum; The frustum also has sides, bushings and shafts.

Frustum and frustum are collectively called frustum.

Step 4: Ball

Taking the straight line with the diameter of the semicircle as the rotation axis, the geometry formed by one rotation of the semicircle surface is called spherical surface, which is called ball for short; The center of the semicircle is called the center of the ball, the radius of the semicircle is called the radius of the ball, and the diameter of the semicircle is called the diameter of the ball.

(5) assembly

Complex geometric shapes composed of cylinders, cones, platforms, balls and other geometric shapes are called components.

2. Three views of space geometry

Three views are spatial geometric figures drawn by observers observing the same geometric figure from different positions.

He specifically includes:

(1) Front view: a projection obtained by projecting an object in the front and rear direction;

It can reflect the height and length of an object;

(2) Side view: the projection obtained by projecting the object in the left and right direction;

It can reflect the height and width of an object;

(3) Top view: the projection obtained by the vertical projection of the object;

It can reflect the length and width of an object;

3. Intuition of space geometry

(1) diagonal binary mapping method

(1) Establish a rectangular coordinate system, and take OX and OY which are perpendicular to each other from the known horizontally placed plane figure to establish a rectangular coordinate system;

(2) Draw an oblique coordinate system, and draw the corresponding O'X' and O'Y' on the paper (plane) on which the vertical drawing is made, so that ∠ x' o' y' = 45 (or 135), and the planes they determine represent the horizontal plane;

(3) Draw the corresponding figure, in which the line segment parallel to the X axis of the known figure is drawn parallel to the X' axis in the vertical view, and the length is unchanged; The line segment parallel to the Y axis in the known figure is drawn parallel to the Y' axis in the orthographic drawing, and the length becomes half of the original one;

(4) Erase the auxiliary line. After drawing, erase the X axis, Y axis and auxiliary lines (dotted lines) added for drawing.

(2) Parallel projection and central projection

The projection lines of parallel projection are parallel to each other, and the projection lines of central projection intersect at one point.

(2) area and volume

The area and volume formulas of 1. polyhedron

Name Transverse area (S side) Total area (S total) Volume (V)

Sharp ridge

Circumference of straight section of cylindrical prism ×l S side+S bottom H = S straight section H.

Right prism bottom h

Sharp ridge

The sum of the side areas of a cone.

Regular pyramid channel

Sharp ridge

The sum of the areas of each side of the frustum is S side +S upper bottom +S lower bottom h(S upper bottom +S lower bottom+).

Prism (c+c')h'

In the table, s represents the area, c' and c represent the perimeter of the upper and lower bottom surfaces, h represents the inclined height, h' represents the inclined height, and l represents the side length.

2. Formula of area and volume of rotating body

Cylindrical cone truncated spherical body

S side 2πrl πrl π(r 1+r2)l

s all 2πr(l+r)πr(l+r)π(r 1+R2)l+π(r 2 1+R22)4πR2。

vπr2h(πR2L)πR2Hπh(r 2 1+r 1r 2+R22)πR3

In the table, L and H represent the generatrix and height respectively, R represents the bottom radius of cylinder, cone and spherical cap, r 1 and r2 represent the upper and lower bottom radii of frustum respectively, and R represents the radius.

(3) Space points, lines and surfaces

1. Plane overview

Two characteristics of (1) plane: ① infinite extension ② flat (no thickness).

(2) plane drawing: usually draw a parallelogram to represent the plane.

(3) plane representation: use lowercase Greek letters,,, etc. , such as airplanes, airplanes; It is represented by letters representing two opposite vertices of a parallelogram, such as plane AC.

2. Three axioms and three inferences:

Axiom 1: If two points on a straight line are on the same plane, then all points on the straight line are on this plane: A, B, A, B.

Axiom 2: If two planes have a common point, then they have other common points, and the set of all these common points is a straight line passing through this common point.

Axiom 3: After passing through three points that are not on the same straight line, there is one and only one plane.

Inference 1: Through a straight line and a point outside this straight line, there is one and only one plane.

Inference 2: Through two intersecting straight lines, there is one and only one plane.

Inference 3: After passing through two parallel straight lines, there is one and only one plane.

3. Spatial straight line:

The positional relationship between two straight lines in (1) space;

Intersecting straight lines-only one thing in common;

Parallel straight lines-on the same plane, have nothing in common;

Straight lines in different planes-the differences in any plane have nothing in common. Intersecting straight lines and parallel straight lines are also called * * * plane straight lines.

There are three ways to draw straight lines on different planes:

(2) Parallel straight lines:

In plane geometry, two lines parallel to the same line are parallel to each other, and this conclusion also holds in space. Axiom 4: Two lines parallel to the same line are parallel to each other.

(3) Out-of-plane straight line theorem: A straight line connecting a point in the plane with a point out of the plane and an in-plane straight line not passing through the point are out-of-plane straight lines. Inference: AB and A are straight lines on different planes.

4. The positional relationship between a straight line and a plane

(1) The straight line is in the plane (there are many common points);

(2) The straight line intersects the plane (only one common point);

(3) The straight line is parallel to the plane (there is nothing in common)-two classifications are made by dichotomy.

Their graphs can be represented as follows, and their symbols can be represented as,, and so on.

Theorem for judging that a straight line is parallel to a plane: If a straight line not in a plane is parallel to a straight line in a plane, then this straight line is parallel to this plane. Reasoning mode:.

Theorem of parallelism between straight lines and planes: If a straight line is parallel to a plane and the plane passing through it intersects with this plane, then the straight line is parallel to the intersection line. Reasoning mode:.

5. There are two kinds of positional relationships between two planes: two planes intersect (have a common straight line) and two planes are parallel (have no common point).

(1) Theorem for determining the parallelism of two planes: If two intersecting lines in a plane are parallel to one plane, then the two planes are parallel.

Pattern of theorem:

Inference: If two intersecting lines on one plane are parallel to two intersecting lines on another plane, then the two planes are parallel to each other.

Reasoning mode:

(2) The property that two planes are parallel (1) If two planes are parallel, then a straight line on one of them is flat.

6. These lines are vertical.

The method of judging the verticality of a straight line: the angle formed is a right angle, and the two straight lines are vertical; If one of two parallel lines is perpendicular, it must be perpendicular to the other.

Three perpendicularity theorem: if a straight line in a plane is perpendicular to the projection of the plane, it is perpendicular to the diagonal.

Inverse theorem of three perpendicular lines theorem: If a straight line in a plane is perpendicular to a diagonal line of this plane, it is also perpendicular to the projection of this diagonal line.

Reasoning mode:.

Note: (1) The three perpendicular lines mean that PA, PO and AO are all perpendicular to the straight line A within α. Its essence is: the judgment and property theorem that the diagonal is perpendicular to a straight line in the plane. (2) To consider the position of A, pay attention to the alternate use of the two theorems.

7. Lines and planes are perpendicular.

Definition: If the straight line L intersects the plane α and is perpendicular to any straight line in the plane α,

Suppose that the straight line L is perpendicular to the plane α, where the straight line L is called the perpendicular of the plane, the plane α is called the vertical plane of the straight line L, and the intersection of the straight line and the plane is called the vertical foot. The straight line L is perpendicular to the plane α, which is expressed as l⊥α.

Theorem for judging whether a straight line is perpendicular to a plane: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.

Theorem that straight lines are perpendicular to a plane: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.

8. Face to face vertical

Definition of two planes perpendicular: Two planes intersecting into a straight dihedral angle are called mutually perpendicular planes.

Theorem for judging the perpendicularity of two planes: (straight line, plane, vertical plane, vertical plane is perpendicular)

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, the straight line perpendicular to their intersection in one plane is perpendicular to the other plane.

Second, the preliminary analytic geometry

1. Inclination: The minimum positive angle formed by the upward direction of a straight line L and the positive direction of the X axis is called the inclination of the straight line, and the range is.

2. Slope: When the inclination of a straight line is not 90, the tangent value is called the slope of the straight line, that is, k = tan When the inclination of the straight line is equal to 90, the slope of the straight line does not exist.

3. The slope formula of a straight line passing through two points p 1(x 1, y 1) and p2(x2, y2)(x 1≠x2): k=tan (if X 1 = X2,

4. Five forms of linear equation need two independent conditions to determine the linear equation. There are many forms to determine the linear equation of one variable, but we must pay attention to the scope of application of each form of linear equation of one variable.

The name equation explains the applicable conditions.

Oblique formula y = kx+bk- slope

B—— This formula can't be used for straight lines with longitudinal intercept inclination of 90.

Point skew y-y0=k(x-x0) (x0, y0)- on a straight line.

For a known point, k-a straight line with an inclination angle of 90, this formula cannot be used.

Two-point formula = (x 1, y 1), (x2, y2) is a straight line with two known points parallel to two coordinate axes, so this formula cannot be used.

Intercept formula += 1a- transverse intercept of a straight line.

B—— The formula cannot be used for straight lines whose vertical intercept exceeds (0,0) and parallel to the two coordinate axes.

The general formula Ax+By+C=0, which are slope, transverse intercept and longitudinal intercept respectively.

A and b cannot both be zero.

The oblique point and oblique section of a straight line cannot represent a straight line with no slope (perpendicular to the X axis); The two-point formula cannot represent a straight line parallel to or coincident with two coordinate axes; Intercept formula cannot represent the straight line parallel to or coincident with two coordinate axes and the straight line passing through the origin.

5. Parallelism and verticality between L1line and l2 line

(1) If l 1 and l2 all have slopes and do not overlap:

①l 1//L2 k 1 = k2； ②l 1l2 k 1k2=- 1 .

(2) If

If A 1, A2, B 1 and B2 are not zero.

①l 1//L2；

②l 1 L2 a 1 a2+b 1 B2 = 0；

③l 1 intersects with l2;

④l 1 coincides with l2;

Note: If A2 or B2 contains letters, it is important to discuss the case that letters =0 and 0. Intersection of two straight lines: the number of intersections of two straight lines depends on the number of solutions of the equation formed by the equations of these two straight lines.

5. Distance

(1) Distance between two points: If A(x 1, y 1) and B(x2, y2), then

Specifically: shaft, then, shaft, then.

(2) Distance between parallel lines: If, then:. Note: the coefficients of x and y should be equal.

(3) The distance from the point to the straight line:, then the distance from P to L is:

7. Equation of circle

The standard equation of a circle with the center and radius r is: especially the equation of a circle with the center at the origin at that time is:

The general equation of a circle, the center of which is a point and the radius is, where.

Binary quadratic equation, the necessary and sufficient conditions for the equation to represent a circle are: ①. The coefficients of each item are the same and not 0, that is; ② There is no xy term, that is, b = 0；; ③、。

8. There are three relationships between the straight line Ax+By+C=0 and the position of the circle.

(1) If;

(2);

(3)。

You can also use the equation of a straight line and the equation of a circle to solve simultaneous equations, and judge by the number of solutions:

(1) When the equations have two common * * * solutions (lines and circles have two intersections), the lines and circles intersect;

(2) When the equations have and only have 1 common * * * solutions (straight lines and circles have only 1 intersections), the straight lines are tangent to the circle;

(3) When the equations have no common * * * solution (straight lines and circles have no intersection), straight lines and circles are separated;

That is, the linear equation is substituted into the equation of the circle, and the quadratic equation of one variable is obtained. Let the discriminant be δ and the distance from the center of the circle C to the straight line L be d, then the positional relationship between the straight line and the circle satisfies the following relationship:

Tangency d = rδ= 0；;

Intersection d < r δ > 0;

D & gtrδ& lt； 0。

4. Method of judging the positional relationship between two circles

Let the center of two circles be O 1, O 2, and the radius be r 1, r2, O2.

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External separation and external cutting

Intersecting inscribed inclusion

Judging the positional relationship between two circles can also be solved by judging the number of common solutions by simultaneous equations.