Multiplication and factorization
①(a+b)(a-b)= a2-B2； ②(a b)2 = a2 2ab+B2； ③(a+b)(a2-a b+B2)= a3+B3；
④(a-b)(a2+a b+B2)= a3-B3； a2+B2 =(a+b)2-2ab； (a-b)2=(a+b)2-4ab .
Operating characteristics of power supply
①am×an = am+n； ②am÷an = am-n； ③(am)n = amn； ④(ab)n = anbn； ⑤()n =；
⑥ A-n =, especially: ()-n = () n; ⑦a0= 1(a≠0)。
quadratic radical
①()2 = a(a≥0)； ②=?a?； ③=×; ④=(a>0，b≥0).
Triangle inequality
| a |-| b |≤| a b|≤| a |+| b | (theorem);
The reinforcement condition: || a |-| b || ≤| a b |≤| a |+| b | also holds. This inequality can also be called vector triangle inequality (where A and B are vector A and vector B respectively).
| a+b |≤| a |+| b |； | a-b |≤| a |+| b |； | a |≤b & lt； = & gt-b≤a≤b；
| a-b |≥| a |-| b |； -| a |≤a ≤| a |；
The sum of the first n terms of some series.
1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2； 1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2；
2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1)； 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6；
13+23+33+43+53+63+…n3 = N2(n+ 1)2/4； 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3；
monadic quadratic equation
For the equation: ax2+bx+c = 0:
① The formula for finding the root is x =, where △ = b2-4ac is called the discriminant of the root.
When △ > 0, the equation has two unequal real roots;
When △ = 0, the equation has two equal real roots;
When △ < 0, the equation has no real root. Note: When △≥0, the equation has real roots.
② If the equation has two real roots x 1 and x2, the quadratic trinomial AX2+BX+C can be decomposed into a (X-X 1) (X-X2).
(3) A quadratic equation with roots of A and B is x2-(A+B) X+AB = 0.
linear function
The image of the linear function y = kx+b (k ≠ 0) is a straight line (b is the ordinate of the intersection of the straight line and the Y axis, called intercept).
① When k > 0, y increases with the increase of x (straight line rises from left to right);
② When k < 0, y decreases with the increase of x (straight line decreases from left to right);
③ Especially: when b = 0, y = kx (k ≠ 0) is also called proportional function (y is proportional to x), and the image must pass through the origin.
inverse proportion function
The image with inverse proportional function y = (k ≠ 0) is called hyperbola.
(1) When k > 0, the hyperbola is in one or three quadrants (in each quadrant, it descends from left to right);
② When k < 0, hyperbola is in the second and fourth quadrants (in each quadrant, it rises from left to right).
quadratic function
(1). Definition: Generally speaking, if it is a constant, it is called a quadratic function of.
(2) Three elements of parabola: opening direction, symmetry axis and vertex.
The symbol of ① determines the opening direction of parabola: at this time, the opening is upward; At that time, the opening was downward;
Equal, the opening size and shape of parabola are the same.
② Record a straight line parallel to the axis (or coincident). In particular, the axis is recorded as a straight line.
(3) The image features of several special quadratic functions are as follows:
Vertex coordinates of symmetry axis in the opening direction of analytic function
at that time
exploitation
at that time
Opening Down (Axis) (0,0)
(Axis) (0,)
(,0)
(,)
()
(4) Solving the vertex and symmetry axis of parabola.
① Formula method: ∴ The vertex is and the symmetry axis is a straight line.
② Matching method: Using formula method, the analytical expression of parabola is transformed into a form, the vertex is (,) and the symmetry axis is a straight line.
③ Using the symmetry of parabola: Because parabola is an axisymmetric figure with symmetry axis, the intersection of symmetry axis and parabola is the vertex.
If two points on the parabola are known (and the value of y is the same), the equation of symmetry axis can be expressed as:
(5) The function of parabola.
① Determine the opening direction and opening size, which are exactly the same as in.
(2) and * * * both determine the position of the parabola axis of symmetry. Because the parabola axis of symmetry is a straight line.
So: ①, the symmetry axis is the axis; (2) (that is, the symbols are the same), and the symmetry axis is on the left side of the shaft; (3) (that is, the symbols are different), and the axis of symmetry is on the right side of the axis.
③ The size determines the position where the parabola intersects the axis.
At that time, ∴ parabola and axis have only one intersection (0,):
(1), parabola passing through the origin; (2), the positive semi-axis intersecting the shaft; ③ The axis intersects with the negative half axis.
The above three points are still valid when the conclusions and conditions are exchanged. If the symmetry axis of a parabola is on the right side of the axis, then.
(6). Find the analytic expression of quadratic function by undetermined coefficient method.
① General formula: Given three points or three pairs of values on an image, the general formula is usually selected.
② Vertex: The vertex or symmetry axis of the image is known, and the vertex is usually selected.
③ Intersection point: the coordinates of the intersection point between the image and the axis are known, and the intersection point is usually selected.
(7). Intersection point of straight line and parabola
(1) the intersection of axis and parabola is (0,).
(2) The intersection of parabola and axis.
The abscissa of the two intersection points between the quadratic function image and the axis corresponds to the quadratic equation of one variable.
The intersection of parabola and axis can be judged by the discriminant corresponding to the root of quadratic equation in one variable:
A has two intersections () parabola intersects the axis;
B has an intersection (the vertex is on the axis) () The parabola is tangent to the axis;
C has no intersection () and the parabola is separated from the axis.
(3) the intersection of a straight line parallel to the axis and a parabola.
Like ②, there may be 0 intersections, 1 intersection and 2 intersections. When there are two intersections, the vertical coordinates of the two intersections are equal. If the ordinate is 0, the abscissa is two real roots.
(4) The intersection of the image of the linear function and the image of the quadratic function is determined by the number of solutions of the following equation:
Equation group a has two different solutions and two intersections;
When the system of B equations has only one set of solutions, there is only one intersection with it;
Equation C has no solution and no intersection.
⑤ Distance between two intersections of parabola and axis: If the two intersections of parabola and axis are, then
Preliminary statistics
(1) Concept: ① All the objects to be investigated are called the whole, and each object to be investigated is called an individual. Some individuals extracted from the population are called the sample of the population, and the number of individuals in the sample is called the sample size. ② In a set of data, the number that appears the most (sometimes more than one) is called the mode of this set of data. ③ Arrange a set of data in order of size and put the middle number.
(2) Formula: If there are n numbers x 1, x2, …, xn, then:
(1) the average value is:
② Range: The range of a set of data is reflected by the difference between the maximum value and the minimum value of the data. The difference obtained by this method is called range, that is, range = maximum-minimum;
③ Variance: Variance of data, ..., yes,
Then =
④ Standard deviation: the arithmetic square root of variance.
Standard deviation of data, ...,
Then =
The greater the variance of a set of data, the greater the volatility and instability of this set of data.
Frequency and probability
(1) frequency
Frequency =, the sum of each group of frequencies is equal to the total number, and the sum of each group of frequencies is equal to 1. The area of each small rectangle in the frequency distribution histogram is the frequency of each group.
(2) Probability
(1) If the probability of event A is expressed by p, then 0 ≤ p (a) ≤1;
P (inevitable event) =1; P (impossible event) = 0;
② Understand the meaning of probability in specific situations, and calculate the probability of simple events by enumeration (including list and tree drawing).
③ The frequency of repeated experiments can be regarded as an estimate of the probability of events;
acute triangle
① Let ∠A be any acute angle of △ABC, then ∠A's sine: sinA= =, ∠A's cosine: cosA= =, and ∠A's tangent: tana =. While SIN2A+COS2A = 1.
0 < Sina < 1，0 < COSA < 1，Tana > 0。 The larger ∠ A is, the greater the sine and tangent of ∠A is, but the smaller the cosine value is.
② Complementary angle formula: sin(90? -A)=cosA，cos(90？ -A)= Sina.
③ trigonometric function value of special angle: sin30? =cos60？ =，sin45？ =cos45？ =，sin60？ =cos30？ =,
tan30？ =，tan45？ = 1，tan60？ =。
④ Slope: I = =. Let the inclination angle be α, then I = tan α =.
Sine (cosine) theorem
(1) sine theorem a/sina = b/sinb = c/sinc = 2r; Note: where r represents the radius of the circumscribed circle of the triangle.
The deformation formula of sine theorem: (1) A = 2RSINA, B = 2RSINB, C = 2RSINC(2) Sina: SINB: SINC = A: B: C.
(2) Cosine Theorem B2 = A2+C2-2accosb; a2 = B2+C2-2 bcco sa； C2 = a2+B2-2 ABC OSC；
Note: ∠C is opposite to C, ∠B is opposite to B, ∠A is opposite to A.
formulas of trigonometric functions
Two-angle sum formula
sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-sinb cosa
cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)ctg(A-B)=(ctgActgB+ 1)/(ctg B-ctgA)
Double angle formula
tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA
cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a
half-angle formula
sin(A/2)=√(( 1-cosA)/2)sin(A/2)=-√(( 1-cosA)/2)
cos(A/2)=√(( 1+cosA)/2)cos(A/2)=-√(( 1+cosA)/2)
tan(A/2)=√(( 1-cosA)/(( 1+cosA))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))
ctg(A/2)=√(( 1+cosA)/(( 1-cosA))ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))
Sum difference product
sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2 cosA+cosB = 2 cos((A+B)/2)sin((A-B)/2)
tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb
ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb
Sum and difference of products
2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)
2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = cos(A+B)-cos(A-B)
Related knowledge of plane rectangular coordinate system
(1) symmetry: if there is a point P(a, b) in the rectangular coordinate system, the point p is called P 1(a, -b) with respect to the x pair, P2 (-a, b) with respect to the y pair, and P3 (-a, -b) with respect to the origin pair.
(2) Coordinate translation: if a point P(a, b) in the rectangular coordinate system is translated by H units to the left, the coordinate becomes P (a-h, b), and translated by H units to the right, the coordinate becomes P (a+h, b); If you translate H units up, the coordinates will become P(a, b+h), and if you translate H units down, the coordinates will become P(a, b-h). For example, if point A (2,-1) is shifted up by 2 units and then shifted to the right by 5 units, the coordinate becomes A (7, 1).
Formula of sum of angles in polygons
Formula for the sum of polygon internal angles: the sum of n polygon internal angles is equal to (n-2) 180? (n≥3, n is a positive integer), and the sum of external angles is equal to 360?
Proportional Theorem of Parallel Lines
(1) Proportional theorem of parallel lines: three parallel lines cut two straight lines, and the corresponding line segments are proportional.
As shown in the figure: A∨B∨C, straight lines l 1 and l2 intersect with straight lines A, B, C and points A, B, C and D, E and F respectively.
Yes.
(2) Inference: A straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments obtained are proportional. As shown in the figure: △ABC, where DE∑BC, DE intersects with AB and AC and points D and E, then:
Projective Theorem in Right Triangle
Projection theorem in right triangle: as shown in figure: Rt△ABC, ∠ ACB = 90o, CD⊥AB in D.
Then there are: (1)(2)(3)
Related properties of circle
(1) vertical diameter theorem: if a straight line has any two of the following five properties: ① passing through the center of the circle; ② Vertical chord; ③ bisect the chord; (4) bisecting the lower arc of the chord; (5) bisecting the optimal arc of the chord, then this straight line has three other properties. Note: When ① and ③ are satisfied, the chord cannot be the diameter.
(2) The arcs sandwiched by two parallel chords are equal.
(3) The degree of the central angle is equal to the degree of the arc it faces.
(4) The angle of the circle subtended by the arc is equal to half the angle of the center of the circle subtended by it.
(5) The angle of a circle is equal to half the angle of the arc it faces.
(6) The circumferential angles of the same arc or equal arc are equal.
(7) In the same circle or in the same circle, the circular arcs with equal circumferential angles are equal.
(8)90? The chord subtended by the circumferential angle is the diameter, and the circumferential angle subtended by the diameter is 90 degrees. The diameter is the longest chord. 、
(9) Diagonal complementation of quadrilateral inscribed in a circle.
Inner and outer center of triangle
(1) The center of the inscribed circle of a triangle is called the interior of the triangle. The interior of a triangle is the intersection of three internal bisectors.
(2) The center of the circumscribed circle of a triangle is called the outer center of the triangle. The outer center of a triangle is the intersection of the vertical lines of three sides.
Common conclusions: ① ① The three sides of RT △ ABC are: A, B and c(c is the hypotenuse), which is the radius of its inscribed circle;
② The circumference of △ ABC is, the area is S, and the radius of its inscribed circle is R, then
Chord tangent angle theorem and its inference
(1) Chord angle: the angle whose vertex is on the circle, one side intersects the circle and the other side is tangent to the circle is called chord angle. As shown in the figure: ∠PAC is the chord tangent angle.
(2) Chord angle theorem: the degree of chord angle is equal to half of the degree of arc it encloses.
If AC is the chord of ⊙O, PA is the tangent of ⊙O and A is the tangent point, then
Inference: The chord tangent angle is equal to the circumferential angle of the clamped arc (the function proves that the angles are equal).
If AC is the chord of ⊙O, PA is the tangent of ⊙O and A is the tangent point, then
Intersecting chord theorem, secant theorem and secant theorem
(1) Intersecting chord theorem: Two chords intersect in a circle, and the product of the length of two straight lines divided by the intersection point is equal.
As shown in Figure ①, namely: PA PB = PC PD.
(2) Secant theorem: Two secant lines of a circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant line and the circle is equal. As shown in Figure ②, namely: PA PB = PC PD.
(3) Tangent Theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines from this point to the intersection of the secant and the circle. As shown in Figure ③, namely: PC2 = PA PB.
① ② ③
Area formula
①S is positive△ =× (side length) 2.
②S parallelogram = base × height.
③S rhombus = base× height =× (diagonal product),
④
⑤S circle = π R2.
⑥l circumference = 2π r.
⑦ Arc length l =.
⑧
⑨S cylindrical edge = bottom circumference × height = =2πrh,
S total area = S side +S bottom = 2π RH+2π R2
Attending s cone edge =× bottom circumference× bus = π Rb,
S total area = S side +S bottom = π Rb+π R2
Chapter 14 Graphic Similarity
Test center 1, proportional line segment (3 points)
1 and related concepts of proportional line segment
If two line segments A and B are measured in the same length unit, and the length of each line segment is M and N, then the ratio of these two line segments is, or written as A: B = M: N..
In the ratio A: B of two line segments, A is called the first term of the ratio, and B is called the last term of the ratio.
In four line segments, if the ratio of two of them is equal to the ratio of the other two line segments, then these four line segments are called proportional line segments.
If four A, B, C and D satisfy or A: B = C: D, then A, B, C and D are called proportional terms, line segments A and D are called out-of-proportion terms, line segments B and C are called in-proportion terms, and line segment D is called the fourth proportional term of A, B and C.
If there are two identical line segments as items in the proportion, that is, a: b = b: c, then line segment B is called the proportional median of line segments A and C..
2, the nature of the proportion
Basic properties of (1)
①a:b=c: Dad =bc
②a:b=b:c
(2) Comparative nature (internal or external exchange rate)
(exchange content)
(Exchange external projects)
(Exchange internal and external projects at the same time)
(3) Inverse ratio (ratio of exchange conditions before and after):
(4) Comprehensive performance:
(5) Equidistant attribute:
3, the golden section
The AB line is divided into AC and BC lines (AC >；; BC), and let AC be the average of the ratio of AB to BC, which is called the golden section point of line segment AB, and point C is called the golden section point of line segment AB, where AC=AB0.6 18AB.
Test center 2. Proportional Theorem of Parallel Lines (3~5 points)
Three parallel lines cut two straight lines, and the corresponding line segments are proportional.
Inference:
(1) A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the corresponding line segment is proportional.
Inverse theorem: If the corresponding line segments obtained by cutting two sides of a triangle (or the extension lines of two sides) are proportional, then this straight line is parallel to the third side of the triangle.
(2) The three sides of the triangle cut by the line parallel to one side of the triangle and intersecting with the other two sides are in direct proportion to the three sides of the original triangle.
Test center three, similar triangles (3~8 points)
1, the concept of similar triangles
A triangle with equal angles and proportional sides is called a similar triangles. Similarity is represented by the symbol ∽, which is pronounced as "similar to". The ratio of similar triangles to corresponding edges is called similarity ratio (or similarity coefficient).
Similar triangles's fundamental theorem.
A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides) to form a triangle similar to the original triangle.
Expressed in mathematical language as follows:
∵DE∥BC,∴△ADE∽△ABC
Similar triangles's equivalence relation;
(1) reflexivity: for any △ABC, there is △ ABC ∽△ ABC;
(2) Symmetry: if △ABC∽△A'B'C', △A'B'C'∽△ABC.
(3) transitivity: if △ABC∽△A'B'C' and △A'B'C', △ ABC ∽△ A' b' c'.
3. Judgment of triangle similarity
The Method of Judging the Similarity of (1) Triangle
① Definition method: Two triangles with equal corresponding angles and proportional corresponding sides are similar.
② Parallel method: A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.
③ Decision Theorem 1: If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar, which can be simply described as two angles are equal and two triangles are similar.
④ Decision Theorem 2: If two sides of a triangle correspond to two sides of another triangle equally, and the included angle is equal, then the two triangles are similar, which can be simply described as two sides corresponding in proportion and the included angle is equal, and the two triangles are similar.
⑤ Decision Theorem 3: If three sides of a triangle are proportional to three sides of another triangle, then two triangles are similar, which can be simply described as three sides are proportional and two triangles are similar.
(2) The method of judging the similarity of right triangle.
① The above judgment methods are all applicable.
(2) Theorem: If the hypotenuse and a right-angled side of a right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.
③ Vertical method: two right-angled triangles divided by the height on the hypotenuse are similar to the original triangle.
4. The nature of similar triangles
(1) similar triangles have equal angles and proportional sides.
(2) The ratio of the corresponding height of similar triangles, the ratio of the corresponding midline and the ratio of the corresponding angular bisector are all equal to the similarity ratio.
(3) The ratio of similar triangles perimeter is equal to the similarity ratio.
(4) The ratio of similar triangles area is equal to the square of similarity ratio.
5. Similar polygons
(1) If the angles of two polygons with the same number of sides are equal and the corresponding sides are proportional, then the two polygons are called similar polygons. The ratio of the corresponding edges of similar polygons is called similarity ratio (or similarity coefficient).
(2) Properties of similar polygons
① The angles of similar polygons are equal, and the corresponding edges are proportional.
② The ratio of the perimeters of similar polygons and the ratio of the corresponding diagonals are equal to the similarity ratio.
③ The corresponding triangles in similar polygons are similar, and the similarity ratio is equal to similar polygons.
④ The area ratio of similar polygons is equal to the square of the similarity ratio.
6, like graphics
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, and this point is called potential center, and the similarity ratio at this time is called potential ratio.
Property: each group of corresponding points is on the same straight line with the potential center, and the ratio of their distance to the potential center is equal to the potential ratio.
The transformation from a graph to its potential graph is called potential transformation. A figure can be enlarged or reduced by potential transformation.