How to express the properties of the perpendicular bisector in symbolic language

Vertical bisector.

It refers to a straight line that passes through the midpoint of a certain line segment and is perpendicular to this line segment, which is called the perpendicular bisector (perpendicular line) of the line segment.

Nature of the perpendicular bisector:

2.Any point on the perpendicular bisector is equal at the same distance from both ends of the segment.

3.If two shapes are about a straight line pair.

Known: AOB + AOC = 180°

OD and OE divide AOB and AOC equally, respectively

Verification: doe=90°

Proof: OD divides AOB equally

aod=1/2∠aob

Similarly. aoe=1/2∠aoc

doe=∠aod+∠aoe

1/2∠aob+1/2∠aoc

1/2(∠aob+∠aoc)

So doe=90°

Therefore, the bisector lines that are adjacent to each other are perpendicular to each other.

The segment is perpendicular and bisected, and the trajectory to a point at which the two ends of the segment are equally apart is the perpendicular bisector of the segment.

Theorem of the properties of the perpendicular bisector:

2. At any point on the perpendicular bisector line, the distance from the two ends of the line segment is equal.

3. The perpendicular bisector of the three sides of the triangle intersects at a point, which is called the outer center, and the distance from this point to the three vertices is equal.

4. The straight line that passes through the midpoint of a certain line segment and is perpendicular to this line segment is called the vertical bisector of this line segment, also known as the "middle perpendicular line".

Extended Materials. 1. Median line, the connection line of any two midpoints on any two sides of the triangle is called the median line. It is parallel to the third side and equal to half of the third side.

2. Height, draw a perpendicular line from a vertex to the straight line where its opposite edge is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.

3. Angle bisector, the bisector of an inner angle of the triangle intersects with the opposite side of the angle, and the line segment between the vertex and the intersection point of the angle is called the angle bisector of the triangle.

4. The middle line, the line segment that connects a vertex of the triangle and the midpoint of the opposite side is called the center line of the triangle.

The perpendicular bisector symbol is . , which is a symbol that indicates perpendicularity, that is, two line segments are perpendicular to each other at an angle of 90 degrees. Rather than a perpendicular bisection, a perpendicular bisector is a line segment perpendicular to another line segment and splitting the other line slippage into two parts.

A trajectory that perpendicularly and bisects a line segment to a point at equal distances from the two endpoints of the line segment.

is the perpendicular bisector of this line segment.

You can also directly write the four words vertical decide, and the standard answer for the exam is like this.

Definition of the bisector

A straight line that passes through the midpoint of a certain line segment and is perpendicular to this line segment is called the perpendicular bisector of the line segment, also known as the "middle perpendicular line."

The perpendicular bisector can be seen as a set of points at equal distances from the two endpoints of the line segment, and the perpendicular bisector is an axis of symmetry of the line segment.

It is a very important part of the geometry discipline of Chuqiao University and Middle School. A bisector divides a line segment into two equal segments from the middle and at a 90-degree angle perpendicular to the dividing segment. A straight line that passes through the midpoint of a certain line segment and is perpendicular to this line segment is called the perpendicular bisector of the line segment, also known as the "middle perpendicular line."

The symbolic language of the theorem of the perpendicular nature of the line and surface is " " to indicate a vertical relationship, " to indicate a relationship ", " to indicate a relationship of irregularity, " to indicate an intersection relationship, " to indicate arbitrary.

Line-surface perpendicularity means that in three-dimensional space, when a straight line intersects a plane, the point where the intersection of the line and the plane is located is inside the plane, and all the points on the line are perpendicular to the intersection line on the plane. It can be represented by the symbol "l p", where l is a straight line and p is a plane.

Let the line l and the plane p intersect at the point a. then l is perpendicular to p and denoted as l p if and only if the following conditions are met: l is inside p, i.e., l and p have a common point a; Any point on l is perpendicular to any straight line on p that passes through point a, and is expressed in symbolic language as:

l ⊥ p ⇔ a ∈ l ∩ p, ∀p∈ p, a,l⊥ p。where " denotes a vertical relationship, " denotes a relationship that belongs, " denotes an intersection relationship, and " denotes arbitrary.

The property theorem of perpendicular lines-planes is one of the fundamental properties in geometry and is used to describe perpendicular relationships. This theorem is one of the fundamental theorems in Euclidean geometry, which expresses the vertical relationship between a straight line and a plane. The theorem can be expressed succinctly in symbolic language, which avoids ambiguity and imprecision in natural language expressions.

To prove that a line is perpendicular to a plane, two conditions need to be met at the same time: the line is inside the plane, and any point on the line is perpendicular to any line on the plane that passes through the intersection. The state's stuffy theorem can also be generalized to the vertical relationships of points, lines, and planes in space, and can be expressed by the calculation of vectors and point products.

This theorem has a wide range of practical applications, such as architectural design, machining, geographic surveying and other fields, all of which need to use this theorem to calculate vertical relationships.

The application of the perpendicular nature of the line surface

Intuitively, if we think of the plane p as a table, then a wooden stick standing on the table is a straight line perpendicular to the tabletop. The vertical property of the line plane is a basic property in Euclidean geometry, which can be generalized to the vertical relationship between any two straight lines or two planes in three-dimensional space. This property has a wide range of applications in geometry, especially in the fields of architecture, mechanical engineering, and geographic surveying.

a⊥m，a⊥n，m∩n＝a，mα，nαa⊥α。

Analysis: If a straight line is perpendicular to two intersecting lines in a plane, the line is perpendicular to the plane. Proof: Known: Straight Line, Verification: A Plane.

Proof that if p is any straight line in the plane, then only a p is required, and the direction vectors of the lines a, b, c, and p are respectively proved, and it is only necessary to prove that b and c are not colinear, and that the lines b, c, and p are in the same plane, and that there are real numbers according to the fundamental theorem of plane vectors, such that the line a is perpendicular to the plane.

The property theorem of the perpendicular nature of lines and surfaces Symbolic language refers to a geometric theorem that lines and surfaces are perpendicular to each other, and its expression can be expressed in a symbolic language. Specifically, for a plane and a line, if the line and any point on the plane are perpendicular to the plane, then the comma line and the plane are perpendicular to the plane.

From another point of view, this theorem indicates the perpendicular relationship between lines and planes, and also provides a way to judge this relationship. This theorem has a wide range of applications in geometry, such as computer graphics, architectural design, and other fields.

It is important to note that this theorem is derived based on the axioms and definitions of Euclidean geometry, so it only holds in Euclidean spaces. For space in non-Euclidean geometry, there may be no vertical relationship, or the definition of the vertical relationship needs to be modified.

In addition, although the formulation of this theorem is very simple and straightforward, there are still some details that need to be paid attention to in the practical application of the sail. For example, it is necessary to clarify the position relationship between lines and points, the direction of the plane and other factors, and at the same time, it needs to be flexibly applied according to the specific situation to ensure accuracy and practicability.

Therefore, when using this theorem, it is not only necessary to master the relevant symbolic language, but also to make appropriate adjustments and applications according to the actual situation, so as to better exert its application value. <>

Symbolic language: because a, b, a Liang friend b, so a. The imitation theorem implies that a straight line outside the plane is parallel to a straight line inside the plane. The line is parallel to the plane.

Line-plane parallel determination theorem1. If a straight line outside the plane is parallel to a straight line in this plane, the straight line is parallel to this plane.

2. A straight line outside the plane is perpendicular to the perpendicular line of this plane, then this straight line is parallel to this plane.

Line and surface parallel judgment method1) Utilization definition: prove that there is no common point between the straight line and the plane;

2) Using the decision theorem: from the straight line parallel to the straight line, the straight line is parallel to the plane;

3) Take advantage of the property of parallel surfaces: if two planes are parallel, then the straight lines in one plane must be parallel to the other.

Note: Line and plane parallelism is usually verified by constructing a parallelogram.

The determination theorem of parallel facesThe straight lines a and b are both in the plane, and a b = a, a , b, then the symbolic language is: a, b, a b=a, a, b

Known: AOB + AOC = 180°

OD and OE divide AOB and AOC equally, respectively

Verification: doe=90°

Proof: OD divides AOB equally

aod=1/2∠aob

The same goes for AOE=1 2 AOC

doe=∠aod+∠aoe

1/2∠aob+1/2∠aoc

1/2(∠aob+∠aoc)

So doe=90°

Therefore, the bisector lines that are adjacent to each other are perpendicular to each other.

The nature of the perpendicular bisector of a line segment.

Symbolic language. Mn bisects ab, Pa=Pb. perpendicularly