The third grade mathematics knowledge points Book 1 1
Special parallelogram
1 and the properties and judgment of diamond
(1) Definition of diamond:
A set of parallelograms with equal adjacent sides is called a diamond.
(2) the nature of the diamond:
It has the property of parallelogram, and its four sides are equal. Two diagonals are bisected vertically, and each diagonal bisects a set of diagonals.
The diamond is an axisymmetric figure, and the straight line where each diagonal line is located is the axis of symmetry.
(3) Diamond identification method:
A set of parallelograms with equal adjacent sides is a diamond.
Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond.
2. The nature and judgment of rectangle
Definition of (1) rectangle:
A parallelogram with a right angle is called a rectangle. Rectangle is a special parallelogram.
② The nature of rectangle:
It has the nature of a parallelogram with equal diagonals and four right angles. A rectangle is an axisymmetric figure with two axes of symmetry. )
③ Determination of rectangle:
A parallelogram with a right angle is called a rectangle (by definition).
A parallelogram with equal diagonal lines is a rectangle.
A quadrilateral with four equal angles is a rectangle.
④ Inference: The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
3, the nature and judgment of the square
Definition of (1) square:
A set of rectangles with equal adjacent sides is called a square.
(2) the nature of the square:
A square has all the properties of parallelogram, rectangle and diamond. A square is an axisymmetric figure with two axes of symmetry.
(3) Square common judgment:
A diamond with a right angle is a square;
A rectangle with equal adjacent sides is a square;
The rhombus with equal diagonal lines is a square;
A rectangle with diagonal lines perpendicular to each other is a square.
④ The relationship among squares, rectangles, diamonds and parallel sides.
⑤ Trapezoidal definition:
A quadrilateral whose opposite sides are parallel but not parallel is called a trapezoid.
Two trapezoid with equal waist are called isosceles trapezoid.
A trapezoid with a vertical waist bottom is called a right-angled trapezoid.
6. The nature of the isosceles trapezoid:
The two internal angles on the same bottom of the isosceles trapezoid are equal to the diagonal.
Two trapeziums with equal internal angles on the same base are isosceles trapeziums.
The center line of the triangle is parallel to the third side and equal to half of the third side.
The parallel segments sandwiched between two parallel lines are equal.
In a right triangle, the center line of the hypotenuse is equal to half of the hypotenuse.
Mathematics knowledge points in the first volume of the third grade
monadic quadratic equation
1, understand the quadratic equation of one variable
An integral equation containing only one unknown number, and all of them can be reduced to ax2+bx+c=0.
(a, b, c are constants, a≠0), and such an equation is called a quadratic equation.
Let ax2+bx+c=0(a, B, C are constants, and a≠0) be called the general form of a quadratic equation, where A is the coefficient of the quadratic term; B is a linear coefficient; C is a constant term.
2. Solve the quadratic equation of one variable by collocation method.
① Matching method:
Basic steps of solving quadratic equation with one variable by matching method;
Turn the equation into the general form of a quadratic equation with one variable;
Convert the quadratic coefficient into1;
Move the constant term to the right of the equation;
Add half the square of the first coefficient on both sides;
Convert the equation into a form;
Seek the roots of both sides.
3. Solve the quadratic equation of one variable by formula method.
(2) Formula method (note that when looking for abc, you must first turn the equation into a general form)
4. Solve the quadratic equation of one variable by factorization.
③ factorization method
One side of the equation becomes 0 and the other side becomes the product of two linear factors to solve it. (mainly including "improving common factor" and "cross multiplication")
5. The relationship between the roots and coefficients of a quadratic equation.
The relationship between (1) root and coefficient;
When b2-4ac >. 0, the equation has two unequal real roots;
When b2-4ac=0, the equation has two equal real roots;
When B2-4ac
② If the two roots of the unary quadratic equation ax2+bx+c=0 are x 1 and x2 respectively, then:
(3) The function of the relationship between the roots and coefficients of a quadratic equation;
Know one equation and find the other;
Solve the equation and find the value of the symmetric formula of the roots of the quadratic equation x 1 and x2, paying special attention to the following formula:
Knowing the two x 1 and x2 of the equation, we can construct an unary quadratic equation:
x2-(x 1+x2)x+x 1x2=0
Knowing the sum and product of two numbers x 1 and x2, the problem of finding these two numbers can be transformed into finding the root of quadratic equation x2-(x1+x2) x+x1x2 = 0.
6, the application of a quadratic equation
① When solving application problems with equations, there are mainly two steps:
Set an unknown number (when setting an unknown number, in most cases, only one question entitled X is asked; But sometimes it must be considered according to known conditions, equivalence relations and other aspects);
Find equivalence relation (a general topic will contain a sentence expressing equivalence relation, and only need to find this sentence to list equations according to it).
② The process of dealing with problems can be further summarized as follows
The third grade, the first volume, mathematical knowledge points 3
Similarity of graphics
1, proportional line segment
(1) the proportion of line segments
If two line segments AB and CD are measured with the same length unit, the length of them is m and n respectively, then the ratio of these two line segments AB and CD is = m: n, or written as
In the four line segments A, B, C and D, if the ratio of A to B is equal to the ratio of C to D, that is
Then these four line segments A, B, C and D are called proportional line segments.
② Note:
A:b=k, which means that a is k times that of B.
Since the lengths of line segments A and B are both positive numbers, k is a positive number.
The ratio has nothing to do with the length unit of the selected line segment, and the length units of the two line segments should be consistent when solving.
Except a=b, a: b ≠ b: a.
The basic nature of proportion: if
Then ad = bc If ad=bc, then
2. Parallel lines are divided into line segments in proportion.
Proportional theorem of parallel lines divided into segments: three parallel lines cut two straight lines, and the corresponding segments are proportional. As shown in figure 2, l1/L2//L3, then
3. golden section
As shown in figure 1, point C divides AB line into AC and BC lines. if
Then the golden section of line segment AB divided by point C is called the golden section of line segment AB, and the ratio of AC to AB is called the golden section ratio.
The golden section is the most beautiful and pleasing point.
4. Similar polygons
① Meaning:
Generally, figures with the same shape are called similar figures.
Two polygons with equal corresponding angles and proportional corresponding sides are called similar polygons. The ratio of corresponding edges of similar polygons is called similarity ratio.
② Note:
Among the similar polygons, similar triangles is the simplest one.
A triangle with equal angles and proportional sides is called similar triangles. The ratio of corresponding edges in similar triangles is called similarity ratio.
Congruent triangles is a special case of similar triangles, when the similarity ratio is equal to 1.
Note: Just like two similar triangles and two congruent triangles, the letters representing the corresponding vertices should be written in the corresponding positions.
Similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.
The ratio of the circumference of similar triangles is equal to the similarity ratio.
The ratio of similar triangles area is equal to the square of similarity ratio.
The perimeter of similar polygons is equal to the similarity ratio; The area ratio is equal to the square of the similarity ratio.
5. Explore the conditions of triangle similarity.
(1) similar triangles's judgment method:
② 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) the proof of similar triangle judgment theorem.
④ Using similar triangles to measure the height.
⑤ The nature of similar triangles.
⑥ Similarity of graphics
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