Chapter 14 Axisymmetric
A straight line that passes through the midpoint of a line segment and is perpendicular to the line segment is called the midline of the line segment.
The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.
The point on the vertical line in a line segment is equal to the distance between the two endpoints of the line segment.
The axisymmetric figure obtained from plane figure is called axisymmetric transformation.
The nature of isosceles triangle;
The two base angles of an isosceles triangle are equal. (equilateral and angular)
The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other. (Three lines in one) (Attached: top angle +2 bottom angle = 180)
If the two angles of a triangle are equal, then the opposite sides of the two angles are equal. (Equiangular and Equilateral)
An isosceles triangle with an angle of 60 is an equilateral triangle.
In a right triangle, if an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse.
Chapter 15 Algebraic Expressions
The product of numbers or letters is called a monomial. A single number or letter is also a monomial.
The numerical factor in a single item is called the coefficient of the item.
In a monomial, the sum of the exponents of all the letters is called the degree of the monomial.
The sum of several monomials is called polynomial. Each monomial is called a polynomial term ($ TERM) and those without letters are called constant terms.
The degree of the term with the highest degree in a polynomial is the degree of this polynomial.
Monomial and polynomial are collectively called algebraic expressions.
Items with the same letter and the same letter index are called similar items.
Merging similar terms in polynomials into one term, that is, adding their coefficients as new coefficients, while the letter part remains unchanged, is called merging similar terms.
The addition and subtraction of several algebraic expressions is usually to enclose each algebraic expression in brackets and then connect them with addition and subtraction signs; Then remove the brackets and merge similar projects.
Same radix power multiplication, constant radix, exponential addition.
Power, constant radix, exponential multiplication
The power of the product is equal to multiplying each factor of the product by the power, and then multiplying it by the power.
Multiply a monomial by a monomial, and multiply them by their coefficients and the same letters respectively. For letters contained only in the monomial, they are used as a factor of the product together with its index.
Multiplying a polynomial by a monomial is to multiply each term of a polynomial by a monomial, and then add the products.
Multiply polynomials by multiplying each term of one polynomial by each term of another polynomial, and then add the products.
(x+p)(x+q)=x^2+(p+q)x+pq
Square difference formula: (a+b) (a-b) = a 2-b 2.
Complete square formula: (a+b) 2 = A2+2ab+B2 (a-b) 2 = A2-2ab+B2.
(a+b+c)^2=a^2+2a(b+c)+(b+c)^2
Same base powers divides, the base remains the same, and the exponent is subtracted.
Any number that is not equal to the power of 0 is equal to 1.
Chapter 16 Scores
If A and B represent two algebraic expressions and B contains letters, then the formula A/B is called a fraction.
The numerator of a fraction is multiplied by the denominator or divided by an algebraic expression that is not equal to 0, and the value of the fraction remains the same.
Law of fractional multiplication: fractional multiplication, the product of molecules is the numerator of the product, and the product of denominator is the denominator.
Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor.
Fractional power should be numerator and denominator respectively.
A-n = 1/A n (A ≠ 0) That is to say, A-n (A ≠ 0) is the reciprocal of A n.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
Chapter 17 Inverse proportional function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The image of inverse proportional function belongs to hyperbola.
When k > 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second quadrant and the fourth quadrant respectively, and the y value of each quadrant increases with the increase of x value.
Chapter 18 Pythagorean Theorem
Pythagorean Theorem: If the lengths of two right-angled sides of a right-angled triangle are A and B respectively and the length of the hypotenuse is C, then A 2+B 2 = C 2.
Inverse Theorem of Pythagorean Theorem: If the lengths of three sides of triangle A, B and C satisfy A 2+B 2 = C 2, then the triangle is a right triangle.
A proposition that is proved to be correct is called a theorem.
We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Pythagorean Theorem Inverse Theorem)
Chapter 19 Quadrilateral
A quadrilateral with two sets of parallel opposite sides is called a parallelogram.
The nature of parallelogram: the opposite sides of parallelogram are equal; Diagonal angles of parallelogram are equal. Diagonal bisection of parallelogram.
Determination of parallelogram;
1. Two groups of quadrangles with equal opposite sides are parallelograms;
2. The quadrilateral whose diagonal lines bisect each other is a parallelogram;
3. Two groups of quadrangles with equal diagonal are parallelograms;
4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
The center line of the triangle is parallel to the third side of the triangle and equal to half of the third side.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided.
Rectangular judgment theorem;
1. A parallelogram with a right angle is called a rectangle.
2. Parallelograms with equal diagonals are rectangles.
A quadrilateral with three right angles is a rectangle.
The nature of the diamond: all four sides of the diamond are equal; The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.
Judgement theorem of diamonds;
1. A set of parallelograms with equal adjacent sides is a diamond.
2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond.
S diamond = 1/2×ab(a and B are two diagonal lines).
The essence of a square: all four sides are equal and all four corners are right angles.
A square is both a rectangle and a diamond.
Square judgment theorem;
1. A rectangle with equal adjacent sides is a square.
Diamonds with right angles are squares.
A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.
The nature of isosceles trapezoid: the two angles on the same base of isosceles trapezoid are equal; The two diagonals of an isosceles trapezoid are equal.
Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid.
The center of gravity of the line segment is the midpoint of the line segment.
The center of gravity of a parallelogram is the intersection of its two diagonals.
The point of doubt when three center lines of a triangle meet is the center of gravity of the triangle.
A rectangle with an aspect ratio of (root number 5- 1)/2 (about 0.6 18) is called a golden rectangle.
Chapter 20 Data Analysis
Arrange a set of data in order from small to large (or from large to small). If the number of data is odd, the middle number is median); This set of data. If the number of data is even, the average of the middle two data is the median of this set of data.
The data with the highest frequency in a set of data is the pattern of this set of data.
The difference between the largest data and the smallest data in a set of data is called the range of this set of data.
The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation and the more stable it is.
Data collection and sorting steps: 1. Collect data. Arrange data 3. Description data 4. Analyze data 5. Write an investigation report 6. Communication 1. mark
1, same base powers division, constant base, exponential subtraction. am an=am-n(a 0)
2. Divide by two monomials, just divide by the coefficient and the same base respectively.
3. The formula in the form (A and B are algebraic expressions, B contains letters, and B 0) is called a fraction. =0(A=0，B 0).
4. Both the numerator and denominator of the fraction are multiplied by (or divided by) the same algebraic expression that is not equal to zero, and the value of the fraction remains unchanged. After simplification, the fraction with no common factor between numerator and denominator is called simplest fraction. The result of fractional operation must be the simplest.
5. The simplest common denominator is the product of the highest power of all factors of each denominator.
6. When a fractional equation is transformed into an integral equation, both sides of the equation are multiplied by an algebraic expression of an unknown number, and the denominator is removed, sometimes a solution (or root) that is not suitable for the original equation may be generated, and this root is called an increased root. Therefore, it is necessary to check when solving the fractional equation.
7. The zeroth power of any number not equal to zero is equal to 1. a0= 1(a 0)
8. The power of -n(n is a positive integer) of any number that is not equal to zero is equal to the reciprocal of the power of n of this number. n=( )n= (a
9. Use scientific notation to represent some numbers with small absolute values, that is, in the form of a, where n is a positive integer, 1 ≤ < 10. For example, 0.00002 1=2. 1
Second and quadratic equation
1 An integral equation with only one unknown and the highest degree of the unknown is 2 is called a quadratic equation with one variable. General form: ax2+bx+c=0(a, B and C are known numbers, where A, B and C are called quadratic coefficient, linear coefficient and constant term respectively.
2. Solution of quadratic equation in one variable: (1) Direct Kaiping method (2) Factorial decomposition method (cross multiplication) (3) Formula method x= (b2-4ac (4) Matching method (see P32 for details).
3. Discriminating formula for roots of quadratic equation with one variable (2-4ac) When a is (1) > 0, the equation has two unequal real roots; (2) When = 0, the equation has two unequal real roots; (3) When < 0, the equation has no real root.
4. The relationship between the root and the coefficient of a quadratic equation (Vieta theorem): ax2+bx+c=0(a, B and C are known numbers. When A≥0, let two equations be x 1, and x2 be X 1+X2 =-, x1.
5. The quadratic equation with x 1 and x2 as roots is:
Cubic and quadratic functions
2. The axis of symmetry of parabola is the axis, and the vertex is the origin. When it is, the opening is upward, and when it is, the opening is downward.
Fourth, the consistency of graphics.
1, two graphs that can completely overlap are congruent graphs. The mutually coincident vertices are called corresponding vertices, the mutually coincident edges are called corresponding edges, and the mutually coincident angles are called corresponding angles.
2. The edges and angles corresponding to congruent graphs are equal.
3. Identification of congruent triangles (1) If three sides of two triangles are equal respectively, then the two triangles are congruent. Note (edge or SSS)(2) If two triangles have two sides and their included angles are equal, the triangles are congruent. Abbreviated as (Angle SAS) (3) Two triangles are congruent if their two angles and their clamping sides are equal respectively, abbreviated as (Angle ASA) (4) If their hypotenuses and a right-angled side are equal respectively, two right-angled triangles are congruent. Abbreviated as (HL)
4. Sentences that can judge whether it is right or wrong are called propositions, which are often written in the form of "if ……", and the original basis for judging the truth of other propositions is called axioms. Some propositions can be judged to be correct by axioms or other true propositions through logical reasoning, and can be further used as the basis for judging the truth value of other propositions. Such a true proposition is called a theorem. According to topics, definitions, axioms, theorems, etc. , through logical reasoning, judge whether a proposition is correct. This reasoning process is called proof.
Verb (abbreviation for verb) circle
1, the related concept of circle: (1), and determine that the elements of a circle are the center and radius. (2) A line segment connecting any two points on a circle is called a chord. The chord passing through the center of the circle is called the diameter. The part between any two points on a circle is called an arc. An arc smaller than half a circle is called a bad arc. An arc larger than half a circle is called an optimal arc. In the same circle or equal circle, arcs that can overlap each other are called equal arcs. The vertex is on the circle, and the angle at which both sides intersect the circle is called the circumferential angle. You can draw a circle through the three vertices of a triangle, and only one can be drawn. The circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle of the triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle is the intersection of the vertical lines of each side of the triangle. The radius of the circumscribed circle of a right triangle is equal to half of the hypotenuse. The circle tangent to each side of the triangle is called the inscribed circle of the triangle, the center of the inscribed circle of the triangle is called the inner circle of the triangle, and this triangle is called the circumscribed triangle. The inner circle of the triangle is the intersection of bisectors of three inner angles of the triangle. The radius of the inscribed circle of a right triangle satisfies:
2. Theorem about the nature of a circle (1) is in the same circle or within the same circle. If the central angles of the circle are equal, then the arcs it faces are equal, the chords it faces are equal, and the chord center distances of the chords it faces are equal. It is inferred that in the same circle or equal circle, if one group of quantities of two central angles, two arcs, two chords or the distance between two chords are equal, then the other groups of quantities of their pairs are equal respectively. (2) Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord. Inference1(i) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects the two arcs opposite to the chord. (Ⅱ) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord. (iii) bisect the diameter of the arc subtended by the chord, bisect the chord vertically and bisect the other arc subtended by the chord. Inference 2 The arcs sandwiched between two parallel chords of a circle are equal. (3) Theorem of circumferential angle: the circumferential angle of an arc is equal to half the central angle of the arc. Inference 1 In the same circle or equal circle, the circumferential angles of the same arc or equal arc are equal, and so are the arcs with equal circumferential angles. Inference 2 The circumferential angles of semicircles or diameters are all equal, all equal to 90. A chord with a circumferential angle of 90 is the diameter of a circle. Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle. (4) Determination and properties of the tangent: Determination theorem: The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle. Property theorem: the tangent of a circle is perpendicular to the radius passing through the tangent point; A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point; A straight line perpendicular to the tangent through the tangent point must pass through the center of the circle. (5) Theorem: Three points that are not on the same straight line determine a circle. (6) The length of the line segment between a point on the tangent of a circle and the tangent point is called the tangent length from the point to the circle; Tangent length theorem: two tangents of a circle can be drawn from a point outside the circle, and their tangents are equal in length. The connecting line between this point and the center of the circle bisects the included angle between the two tangents. (7) The quadrangles inscribed in the circle are diagonally complementary, and one outer angle is equal to the inner diagonal; The sum of the opposite sides of the circumscribed quadrangle is equal; (8) Chord angle theorem: the chord angle is equal to the circumferential angle of the arc pair it clamps. (9) Proportional line segments related to a circle: the theorem of intersecting chords: the product of two intersecting chords in a circle is equal to the length of two lines divided by the intersection. If the chord intersects the diameter vertically, then half of the chord is the proportional average of two line segments formed by its separate diameters. Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the length ratio of the two lines where this point intersects the secant. Draw two secants of a circle 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 and the circle is equal. (10) Two circles are tangent, and the connecting line intersects the tangent point; Two circles intersect, and the connecting line bisects the common chord vertically.
3. Position relation related to the circle
(1) positional relationship between point and circle: point in circle d (2) positional relationship between straight line and circle: straight line and circle are separated (D >；); r)； The straight line is tangent to the circle (), and this straight line is called the tangent of the circle; A straight line intersects a circle (), which is called the secant of the circle. (3) the positional relationship between circles: outward (D >；); r+r)； External cutting; Intersection (); cut()； Contain
4. Calculation within the circle