Chapter 11 Congruent Triangles
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Two triangles with three sides that correspond to their angles are congruent (side by side or SSS)
The triangle's Stability determines that two triangles are congruent when three sides are equal
Two triangles with two sides and their angle counterparts are congruent (side-angle-side or SAS)
Two triangles with two angles and their side counterparts are congruent (angle-side-angle or ASA)
Two triangles with two angles and the opposite side of one of the angles are congruent (angle-angle-angle-side or AAS)
Two triangles with two angles and the opposite side of one of the angles are congruent (angle-angle-side or AAS)
Two triangles with two angles and the opposite side of one of the angles are congruent. /p>
Two right triangles whose hypotenuse and one of the right-angled sides correspond to equal sides are congruent (hypotenuse, right-angled side, or HL)
The point on the bisector of an angle that is equidistant from both sides of the angle
The point in the interior of an angle that is equidistant from both sides of the angle is on the bisector of the angle
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Chapter 12 Axial Symmetry
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Properties of Isosceles Triangles:
Property 1: The two base angles of an isosceles triangle are equal (equilateral sides to equilateral angles)
Property 2: The vertex bisectors of isosceles triangles, and the medians on the base sides, height on the base side coincide with each other.
If a triangle has two angles equal, then the sides opposite those angles are also equal (equal angles to equal sides)
Property of an equilateral triangle:
An equilateral triangle has three interior angles that are equal, and each of those angles is equal to 60 degrees
A triangle with three equal angles is equilateral
An isosceles triangle that has one angle of 60 degrees is equilateral. An isosceles triangle is an equilateral triangle
In a right triangle, if an acute angle is equal to 30 degrees, then the right-hand side it subtends is equal to half of the hypotenuse.
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Chapter 13 Real Numbers
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If the square of a number is equal to a, then the number is called the square root of a or the quadratic root ( square root)
The operation of finding the square root of a number a is called extrapolation of square root
Positive numbers have two square roots, which are opposite to each other.
The square root of 0 is 0
Negative numbers do not have a square root
If the cube of a number is equal to a, then the number is called the cube root or cubic root of a
The operation of finding the cube root of a number is called extraction of cube root
The cube root of a positive number is positive
Positive numbers have two square roots which are opposite to each other. cube root of a positive number
The cube root of a negative number is a negative number
The cube root of 0 is 0
The infinite non-circular decimals are called irrational numbers
Rational and irrational numbers are collectively known as the real numbers
The opposite of the number a is -a
The absolute value of a positive real number is itself, that of a negative real number is its opposite, and that of 0 is 0
A positive real number is itself, and that of a negative real number is the opposite. absolute value is 0
3√a 3 is the root exponent a is the number of squares being opened
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Chapter 14 Primary Functions
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In a change in a process, we call quantities whose values change variable,
some quantities have constant values, and we call them constants
In a process of change, if there are two variables, x and y, and for every determined value of x, there is a uniquely determined value of y that corresponds to it, then we say that x is the
(independent variable), y is a function of x. If y=b when x=a, then b is called the value of the function when the value of the independent variable is a
Three ways to represent a function: tabular, analytic, and pictorial
Proportional function
y=kx (k is a constant, k is not 0) k is a constant of proportionality
The proportional function, the image of which is a straight line passing through the origin, is called the straight line y=kx
When k>0, the straight line y=kx passes through the first and third quadrants (lower left-upper right) and rises from the left to the right, i.e., as x increases, so does y
When k<0, the straight line y=kx passes through the second and fourth quadrants (upper left-bottom right) and descends from left to right, i.e., x increases and y decreases instead
(The proportional function is a straight line that passes through the origin)
(The primary function is a straight line that translates on the y-axis, and this translation is responsible for b in y=kx+b, which is the point of intersection of the straight line with the y-axis)
One-time function
y=kx+ b(k,b is constant,k is not 0) ,primary function (linear function), also as a linear function!
Which b generally represents a function of the change of an initial amount, that is, similar
Existing mileage + speed * time = actual mileage ( y: actual mileage k: time x: speed b: now mileage)
When b = 0, y = kx + b that is, y = kx, that is, the proportional function is a special kind of primary function
To be determined coefficients method
Select two points and write a system of quadratic equations by substituting the coefficients in the format of y=kx+b to solve for the values of k and b.
Any quadratic equation can be converted to the form ax+b=0(a,b are constants, a!=0), i.e.
Solving a quadratic equation, can be understood as finding the corresponding change in the value of the independent variable x in the image of a primary function, y=0
y=kx+b => kx+b=0
In terms of image, it is equivalent to the known straight line y=ax+b and determining the value of the horizontal coordinate of its intersection with the x-axis.
(Find the point of intersection of the x-axis)
Any quadratic inequality can be converted to ax+b>0 or ax+b<0
It can be interpreted as the range of values of the corresponding x-value when the y-value is greater (less) than 0
(There are set representations on the coordinate system in addition to the image.)
Divisional equations (sets) Any of these quadratic equations can be transformed into the form y=kx+b
y varies according to x (and is not limited to =0 <0 >0 in the case of quadratic)
ax+b=0
ax+b<0 or ax+b>0
y=kx+b
The system of two quadratic equations can be transformed into y=kx+b
(The system of coordinates has a set representation in addition to the image)
The system of coordinates has a set representation in addition to the image. A system of two quadratic equations can be understood as the coordinates of the intersection of two lines on a coordinate system
In "numerical" terms, it is the *** same solution to two equations
For example:
System of quadratic equations
3x+5y=8
2x-y = 1
can be evolved into two primary functions (or rather, corresponding to two straight lines)
y = -3/5x + 8/5
y = 2x - 1
Yielding the result that the point of intersection is (1,1)
Generally, each system of quadratic equations corresponds to two primary functions, corresponding to two straight lines.
From a "numerical" point of view, solving the system of equations is equivalent to considering what value of the independent variable is equal to the value of the two functions, as well as what the value of this function is;
From a "formal" point of view, solving the system of equations is equivalent to determining the coordinates of the point of intersection of the two lines. the coordinates of the point of intersection of two lines.
Summary, primary functions and quadratic equations (systems) are closely related
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Chapter 15 Multiplication, Division and Factorization of the Whole Formulas
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15.1 Multiplication of Integers
15.1.1 Multiplying Powers of the Same Base
Multiplying powers of the same base leaves the base unchanged and the exponents are added.
a^n x a^m = a^(m+n)
2^3 x 2^4 = 2^(3+4) = 8x16 = 128 = 2^7
15.1.2
Multiplication of powers, where the base stays the same and the exponents are multiplied.
(a^n)^m = a^(n x m)
15.1.3 Multiplication of a product
Multiplication of a product is the same as multiplying each factor of the product separately, and then multiplying the resulting powers.
(ab)^m = a^mb^m (distributive rate)
15.2 Multiplication formulas
15.2.1 Squared difference formula
(a + b)(a-b) = aa-ab + ab-bb = aa - bb = a^2 - b^2
The product of the sum of two numbers and the difference of those two numbers is equal to the square difference of the numbers The product of the sum of two numbers and the difference between those two numbers is equal to the difference between the squares of those two numbers
Formula for the difference of squares (for multiplication)
15.2.2 The perfect square formula
(a+b)^2 = (a + b)(a+b) = aa + ab + ab + bb = aa+2ab+bb = a^2 + 2ab + b^2
(a-b)^2 = (a-b)(a-b) = aa - ab - ab + bb = aa - ab + bb = aa-2ab + bb = a^2 - 2ab + b^2
The square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product.
When adding parentheses, if the parentheses are preceded by a positive sign, all the items enclosed in the parentheses do not change sign; if the parentheses are preceded by a negative sign, all the items enclosed in the parentheses change sign.
It's the same principle as removing parentheses; it's just a reversal
a+(b+c) = a+b+c
a-(b+c) = a-b-c
15.3 Dividing Whole Forms
15.3.1 Dividing Powers of the Same Base
Dividing powers of the same base leaves the base unchanged, and the exponents subtract from each other.
a^m/a^n = a^(m-n)
The 0th power of any number not equal to 0 is equal to 1.
a^m/a^m = 1
a^(m-m) = 1
a^0 = 1
15.4 Factorization
15.4.1 Raising the Common Factor Method
ma+mb+mc = m(a+b+c)
Formulas Factorization using formulas for integer operations
Negative powers are the inverse of powers a^-n = 1/(a^n)
Also read as a^-n = (a^n)^-1 or (1/a)^n
Positive powers of inverses of the base
First year (next)
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Chapter 16 Fractions
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16.1 Fractions
16.1.1 From Fractions to Fractions
Fractions If the numerator and denominator of a fraction are multiplied (or divided) by an integer not equal to 0, the value of the fraction remains unchanged.
Like the rule for multiplying rational numbers
Fractions that are reduced to approximate fractions and cannot be further divided (have no common factor) are called simplest fractions.
The transformation of two fractions to the same denominator by multiplying them together by a suitable integer is called generalization.
Generally, the product of the highest powers of all factors of each denominator is taken as the common denominator, which is called the simplest common denominator
16.2 Operations on Fractions
Rule of multiplication of fractions: to multiply fractions by fractions, the product of the numerators is taken as the numerator of the product, and the product of the denominators is taken as the denominator of the product.
Rule of division: divide a fraction by a fraction, reverse the numerator and denominator and multiply by the divisor.
Fractional multiplication is done by multiplying the numerator and denominator separately
(a/b)^2 = (a^2)/(b^2) (2 is a square)
To add or subtract fractions with the same denominator, the denominator remains unchanged, and the numerators are added or subtracted.
To add and subtract fractions with dissimilar denominators, pass through the fractions, change them to fractions with the same denominator, then add and subtract.
16.3 Fractional Equations
The idea of solving fractional equations is to turn them into whole equations, which is done by "removing the denominator", i.e. multiplying both sides of the equation by the simplest common denominator, in order to remove the denominator and turn it into a whole
equation.
Generally speaking, when solving a fractional equation, the solution of the integral equation obtained after removing the denominator may make the denominator of the original equation 0, so the test should be as follows:
Substitute the solution of the integral equation into the simplest common denominator, and if the value of the simplest common denominator is not 0, then the solution of the integral equation is the solution of the original fractional equation; otherwise, the solution is not a solution to the original fractional equation
(the original equation has no solution). ).
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Chapter 17 Inverse Proportional Functions
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17.1 Definition of Inverse Proportional Functions
Supplemental Chapter 14 14.2 Notes on Primary Functions
The normal function is y=kx
The primary function is y=kx+b The image is a straight line
The inverse function is y=k/x(k!=0) Hyperbolic (symmetric)
Where x is the independent variable, y is the function, and the range of values of the independent variable, x, is all real numbers not equal to zero. (The denominator cannot be 0)
When k>0, the hyperbolic image is in the first and third quadrants, and the value of y decreases as x increases. (When k>0, x is positive, y is positive i.e., quadrant 1 ,x is negative, y is negative i.e., quadrant 3)
When k<0, the hyperbolic image is in the second and fourth quadrants, the y-value increases as x increases. (k<0, x is positive, y is negative that is 2 quadrant ,x is negative, y is positive that is 4 quadrant)
To determine whether a point is on an inverse function of the same image, first write the inverse function of the analytic formula, and then substituting x, y, to find the constant, the same is on the image!!!!
When you make the images of both positive y=kx+b and inverse y=k/x on the same coordinate system,
it can be seen that the image of the inverse function y=k/x is symmetric about the positive function y=kx as an axis
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Chapter 18 The Pythagorean Theorem
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Proposition 1 If two right-angled sides of a right triangle have lengths a,b and the hypotenuse has length c, then a^2+b^2=c^2
Proposition 2 If three sides of a triangle with lengths a,b,c satisfy a^2+b^2=c^2 . Then the triangle is a right triangle .
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Chapter 19 Quadrilaterals
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19.1 Parallelograms
19.1.1 Properties of Parallelograms
Parallelograms with equal opposite sides
Parallelograms with equal opposite angles
Parallelograms with diagonals bisecting each other
19.1.2 Determination of Parallelograms
Two sets of quadrilaterals with equal opposite sides are parallelograms
Quadrilaterals with opposite sides that bisect each other are parallelograms
A set of Quadrilaterals whose opposite sides are parallel and equal are parallelograms
The median of a triangle is parallel to the third side of the triangle and equal to half of the third side.
19.2 Special Parallelograms
A rectangle has all four angles at right angles
The diagonals of a rectangle are equal
The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
A parallelogram with one angle at right angles is a rectangle.
A parallelogram with equal diagonals is a rectangle.
19.2.2 Rhombus
A parallelogram with a set of equal neighboring sides is called a rhombus
The four sides of a rhombus are equal
The two diagonals of a rhombus are perpendicular to each other and each diagonal bisects a set of opposite angles.
A parallelogram with perpendicular diagonals is a rhombus.
A quadrilateral with four equal sides is a rhombus.
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CHAPTER 20 ANALYSIS OF DATA
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20.1 REPRESENTATION OF DATA
20.1.1 Mean
The mean is the sum of N numbers divided by n. The weighted mean is the sum of the products of N numbers multiplied by their respective weights divided by the sum of the weights of those numbers, which is called the weighted mean
The weights of the data reflect the relative "importance" of the data.
20.1.2 Median and Plurality
A set of data in accordance with the smallest to the largest (or largest to smallest) order, if the number of data is an odd number, the number in the middle of the position is called this set of data in the median (median)
If the number of data is an even number, the middle of the average of the two data is known as this set of data in the median.
The data that occurs most often in a set of data is called the multitude (mode)
If there are two data in a set of data that have the same frequency, both are the maximum, then both are the multitude of the set of data.
20.2 Fluctuations in Data
20.2.1 Extreme Differences
For example, a weather forecast of
Urumqi 24-10 degrees 14 (degrees C)
Guangzhou 25-20 degrees 5 (degrees C)
The difference between these two temperatures shows that on this particular day, the change in temperature in Urumqi was greater, and the change in temperature in Guangzhou was Guangzhou is smaller.
The difference between the largest and smallest data in a set of data is called the range of the data
The range of the data is reflected in the extreme difference.
20.2.2 Variance
The difference between a set of data and its mean is examined to reflect the volatility of the data.
With n pieces of data, multiply the difference between each piece of data and the mean by the square , add the sums and divide by n,
The resulting value is used as a measure of the fluctuation in the set of data, and is called the variance of the set of data, and is denoted as s^2 (s-squared)
s^2 = 1/n [ (x1-x-mean)^2 + (x2-x-mean)^2 + .... + (xn-x mean)^2]
When the data distribution is more dispersed (i.e., the data are larger near the mean), the sum of the squares of the differences of the individual data from the mean is larger, and the variance is larger;
When the data distribution is more concentrated, the sum of the squares of the differences of the individual data from the mean is smaller, and the variance is smaller.
So the larger the variance, the more volatile the data;
the smaller the variance, the less volatile the data.