Knowledge points of mathematics triangle in the second day of junior high school
1, the property theorem and inverse theorem of the vertical line in the line segment
A straight line perpendicular to a line segment and bisecting the line segment is the midline of the line segment.
The property theorem of the median vertical line: the distance between the point on the median vertical line of the line segment and the two endpoints of the line segment is equal. Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment. 2. The bisector of an angle and its properties
A ray divides an angle into two equal angles. This ray is called the bisector of this angle. The bisector of an angle has the following property theorems:
The points on the bisector of the (1) angle are equal in distance to both sides of the angle.
(2) The points with equal distance to both sides of an angle are on the bisector of this angle.
3 the nature of the vertical line:
Property 1: There is one and only one straight line perpendicular to the known straight line.
Property 2: Of all the line segments connecting a point outside and a point on the line, the vertical line segment is the shortest. Abbreviation: the vertical segment is the shortest. 2, the main line segment in the triangle
(1) The bisector of an angle of a triangle intersects the opposite side of the angle, and the line segment between the intersection of the vertex and the angle is called the bisector of the triangle.
(2) In a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the center line of the triangle.
(3) Draw a vertical line from a vertex of a triangle to its opposite side, and the line segment between the vertex and the vertical foot is called the height line of the triangle (the height of the triangle for short).
3. The stability of triangle
The shape of a triangle is fixed, and this property of a triangle is called the stability of a triangle. This property of triangle is widely used in production and life, and things that need stability are generally made into the shape of triangle. 6. Trilateral relation theorem and triangle inference.
(1) Trilateral Relation Theorem: The sum of two sides of a triangle is greater than the third side. Inference: The difference between two sides of a triangle is smaller than the third side.
(2) Trilateral relation theorem of triangle and its inference function;
① Judging whether three known line segments can form a triangle ② When two sides are known, the range of the third side can be determined. ③ Prove the inequality of line segments. 7. Angle relation of triangle
Theorem of sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180. Inference:
① The two acute angles of a right triangle are complementary.
(2) The outer angle of a triangle is equal to the sum of two non-adjacent inner angles. ③ The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
Note: in the same triangle: equilateral and equilateral; Equilateral and angular; Large angle to large side; Large side to large angle. The complementary angles of equal angles are equal, and the complementary angles of equal angles are equal.
8, the area of the triangle
Area of triangle =
2
1
× Bottom× Height Application: The area relationship of two triangles is often used to find the proportional relationship or value of the bottom and height.
Knowledge points of trigonometric proof in eighth grade mathematics
Chapter 1 Proof of Triangle
1, isosceles triangle
Properties and Judgement of (1) Triangular Congruence
The corresponding sides of congruent triangles are equal and the corresponding angles are equal: SSS, SAS, ASA, AAS,
(2) Determination, nature and inference of isosceles triangle.
Properties: The two base angles of an isosceles triangle are equal (equilateral and equiangular).
Judgment: A triangle with two equal angles is an isosceles triangle.
Inference: The bisector of the top angle of an isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide (that is, "the three lines are one")
(3) The properties and judging theorem of equilateral triangle.
Property theorem: all three angles of an equilateral triangle are equal, and each angle is equal to 60 degrees; All three sides of an equilateral triangle meet the property of "three lines in one"; An equilateral triangle is an axisymmetric figure with three axes of symmetry.
Decision Theorem: An isosceles triangle with an angle of 60 degrees is an equilateral triangle. Or a triangle with three equal angles is an equilateral triangle.
(4) The properties of each side of a 30-degree right triangle.
Theorem: In a right triangle, if an acute angle is equal to 30 degrees, then the right side it faces is equal to half of the hypotenuse.
2. Right triangle
(1) Pythagorean Theorem and Its Inverse Theorem
Theorem: The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
Inverse theorem: If the sum of squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.
(2) The relationship between the two acute angles of a right triangle.
Theorem: The two acute angles of a right triangle are complementary.
Inverse theorem: Two triangles with complementary acute angles are right triangles.
(3) The side length theorem of 30-degree right triangle.
Theorem: In a right triangle, if an acute angle is equal to 30 degrees, then the right side it faces is equal to half of the hypotenuse.
Inverse theorem: In a right triangle, a right-angled side is half of the hypotenuse, so the acute angle of this right-angled side is 30 degrees.
Mathematics knowledge points in Grade Two
The concept of similar items: items with the same letters and the same letter index are called similar items. Several constant terms are also called similar terms.
Two criteria for judging whether several monomials or terms are similar;
(1) contains the same letters. The same letter has the same number of times.
When judging similar items, it has nothing to do with coefficient and alphabetical order.
The concept of merging similar terms: merging similar terms in polynomials into one term is called merging similar terms.
Rules for merging similar items: when the coefficients of similar items are added, the result will be taken as the coefficient, and the index of letters will remain unchanged.
To merge similar projects:
(1). Find similar projects accurately.
(2) Reverse the distribution law, add the coefficients of similar items together (enclosed in brackets), and keep the letters and their indices unchanged.
(3) Write the merged result.
Please note when merging similar projects:
(1) If the coefficients of two similar items are opposite, the result after merging similar items is 0.
(2) Don't leave out items that can't be merged.
(3) As long as there are no more similar terms, it is the result (either a monomial or a polynomial).
(4) Projects that do not belong to the same category shall not be merged.
20 17 Senior Two Mathematics Knowledge Points (2)
I concepts of average, median and mode
1. Average
The average value refers to the sum of all data in a set of data divided by the number of data.
2. Median
Median refers to arranging the variable values in the statistical population in order of size to form a sequence, and the variable values in the middle of the variable sequence are called median.
Step 3: Ways
Mode is the value with the highest frequency in a set of data, which is called mode. Sometimes there are several patterns in a set of numbers.
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