geometry is a branch of mathematics which is responsible for studying the properties and relationships of figures in space. One of the fundamental concepts in geometry are bisectors, which are lines that divide an angle into two equal parts. In this article, we will explore in detail what a bisector is, its properties, and examples of how they are used in solving geometric problems.
Bisectors are widely used in geometry for solving construction problems and calculating angles. In addition, the bisectors have applications in everyday life, such as in the design and construction of buildings and bridges.
In this article, we will look at the symbol that is used for represent a bisector, as well as the most relevant properties of this geometric tool. Also, we will describe some practical examples of how bisectors are used in geometry problems and how they are applied in everyday situations.
If you are interested in learning more about geometry and its fundamental concepts, or if you need solve geometric problems In your daily life, this article will provide you with valuable information about bisectors and their application in solving geometric problems.
Concept and definition of a bisector in mathematics
In mathematics, a bisector is a line that divides an angle into two equal parts. This means that the bisector passes through the vertex of angle and divide the sides of the angle into two segments of equal length.
The bisector is a useful tool for the geometric problem solving and is widely used in the construction of angles and triangles. In addition, bisectors also have applications in everyday life, such as in the construction of buildings and bridges.
The concept of bisector is derived from the Latin term ‘bisector’, which means ‘to divide into two equal parts’. In geometry, bisectors are a fundamental tool for problem solving and the construction of geometric figures.
What is the bisector of an angle?
The angle bisector is a line that divides an angle in two equal parts. The bisector passes through the vertex of the angle and divides the sides of the angle into two segments of equal length.
The bisector of an angle can be constructed using a ruler and compass. First, the angle with the extended sides is drawn. Then, the point of the compass is placed at the vertex of the angle and an arc is drawn that intersects both sides of the angle.
Finally, a straight line is drawn passes through the vertex of the angle and by the point where the arc intersects one of the sides of the angle. This line is the angle bisector.
The bisector is a useful tool in the geometric problem solving that involve angles, since it allows calculating the measure of an angle and constructing angles with specific measures.
What is the bisector of a triangle?
The bisector of a triangle is a line that divides one of the angles of the triangle into two equal pairs. bisector of a triangle it can be internal or external.
The internal bisector of a triangle is a line that part from the vertex of the angle and extends to the opposite side, dividing the angle into two equal parts. Every triangle has three internal bisectors, one for each angle.
The external bisector of a triangle is a line that starts from the vertex of the angle and extends towards the exterior of the triangle, so that it forms an angle congruent with half the angle of the vertex. Every triangle has three external bisectors, one for each vertex.
Bisectors are used for the geometric problem solving that involve triangles, as they allow you to build triangles with specific measures and find important points inside the triangle, such as the incenter and excenter.
What are the properties of a bisector?
Here are some important properties of a bisector:
- Divide an angle into two equal parts: The main property of a bisector is that it divides an angle into two equal parts. This means that if we bisector an angle, each of the resulting parts will have half the measure of the original angle.
- Passes through the vertex of the angle: the bisector always passes through the vertex of the angle it is dividing. This means that the bisector is a common point between the two line segments being divided.
- It is perpendicular to the opposite side in the case of an internal bisector: If we draw an internal bisector in a triangle, it will be perpendicular to the opposite side. This means that the point where the bisector intersects the opposite side is the same distance from the other two sides of the triangle.
- Divide the opposite side proportionally in the case of a bisector external: If we draw an external bisector in a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle. This means that if we call x and y the two sides of the triangle that do not contain the vertex where the external bisector is drawn, the segment of the opposite side that is closest to the vertex will have a length of x/(x+y) times the length of the side complete opposite, and the segment that is furthest from the vertex will have a length of y/(x+y) times the length of the complete opposite side.
- The incenter of the triangle is the point where the three internal bisectors intersect.: If we draw the three internal bisectors of a triangle, they will intersect at a common point called the incenter. This point is equidistant from the three sides of the triangle, which means that if we draw a circle with its center at the incenter, this circle will touch each of the sides of the triangle at a single point.
- The three excenters of the triangle are the points where an external bisector and the perpendicular to the opposite side intersect: if we draw the three external bisectors of a triangle, these will intersect the three perpendicular lines to the opposite sides. The points of intersection of each external bisector with the corresponding perpendicular line are called excenters of the triangle. These points are also equidistant from the sides of the triangle, which means that if we draw a circle centered on one of the eccenters, this circle will touch two sides of the triangle and extend to the third side.
What is the symbol for the bisector?
In mathematics, the symbol used to represent the bisector of an angle is a vertical line with a letter ‘B’ inside, which is placed at the vertex of the angle. Therefore, if an angle has a vertex at point A, its bisector can be represented as the line AB with the letter ‘B’ inside the angle.
How is the bisector used?
The bisector of an angle is used in mathematics and in various practical applications. Some common uses of bisector are:
- to build angles: The bisector is used to divide an angle into two equal parts. This is useful for constructing precise angles in geometry.
- in trigonometry: The bisector is useful for determining the measure of an angle, which is important in trigonometry and in solving problems involving angles.
- in graphic design: The bisector is used in graphic design and in creating balanced asymmetrical designs.
- in engineering: The bisector is used in engineering to construct and measure precise angles in various applications.
The bisector is a very useful tool in geometry that is used to divide an angle into two equal parts. Next, We will explain how the bisector is used:
- Draw the angle you want to bisect: First, draw the angle you want to bisect on a piece of paper or drawing software. The angle must be well defined and have a clear vertex.
- Place the tip of the compass at the vertex of the angle: The next step is to place the tip of the compass at the vertex of the angle you want to bisect. If you don’t have a compass, you can use a ruler and pencil to draw two arcs that intersect at the vertex.
- draw an arc: With the compass or ruler and pencil, draw an arc that intersects the two sides of the angle.
- Repeat the previous step: Now, place the point of the compass at the point where the arc intersects one of the sides of the angle and draw another arc that cuts the other side.
- draw the bisector: Finally, draw a straight line that passes through the vertex of the angle and divides the angle into two equal parts. This line is called the bisector of the angle.
The bisector is a very useful tool in geometry and in various practical applications where it is required to divide angles precisely.
Bisector Examples in Mathematics and Geometry
Here we present examples of the application of the bisector in mathematics and geometry:
- construction of angles: Suppose we want to build a 90 degree angle. To do this, we draw an angle of 180 degrees and bisect it with a line that passes through the vertex. In this way, we obtain two angles of 90 degrees each.
- resolution of triangles: In any triangle, the bisector of the angle opposite a side divides that side into two segments proportional to the other two sides of the triangle. This property is used to solve problems involving triangles, such as finding the height of a triangle.
- Graphic design: In graphic design, the bisector is used to create balanced and asymmetric designs. For example, if we want to design a logo, we can use the bisector to divide the image into two equal parts and create a balanced design.
