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"For the next generation"
© 2025 Astra.si. All rights reserved.
"For the next generation"
Plane geometry is a branch of mathematics that studies the properties and relationships between various geometric objects in a two-dimensional space. The basic elements of plane geometry include points, lines, angles, polygons, and circles. These objects have specific properties and laws that can be described using mathematical rules and formulas.
A point is the fundamental building block of geometry, having no dimensions and representing a location in the plane. A line is an infinite collection of points extending in one direction. A line segment is a part of a line bounded by two points, while a ray is a part of a line that has a starting point and extends infinitely in one direction.
Angles are geometric objects formed by two rays sharing a common endpoint (vertex). Based on their size, we distinguish:
A polygon is a geometric figure bounded by several straight sides. The most common polygons include triangles, quadrilaterals, pentagons, and other figures with more sides. Triangles are classified based on the length of their sides (equilateral, isosceles, and scalene) or based on their angles (right-angled, acute-angled, and obtuse-angled). A right-angled triangle has one angle equal to 90 degrees, for which the Pythagorean theorem applies, defining the relationship between the lengths of its sides.
Among quadrilaterals, we distinguish parallelograms, rectangles, squares, and trapezoids, which have different properties regarding the parallelism and length of sides and the size of angles. A square is a special type of rectangle in which all sides are equal in length, and a rectangle has opposite sides of equal length and all angles equal (90 degrees).
A circle is the set of all points in a plane that are equidistant from a central point. The distance from the center to any point on the circle is called the radius. The longest distance through the center is the diameter, which is equal to twice the radius. A chord is a line segment connecting two points on the circle, and an arc is a part of the circle between two points.
A central angle is an angle whose vertex is at the center of the circle, and its corresponding arc has a length proportional to the size of the angle. In circle geometry, the inscribed angle is also often used; it has its vertex on the circle and is equal to half the measure of its intercepted (central) arc.
Plane geometry can also be addressed using coordinate geometry, where the position of points is determined using a coordinate system. The distance between two points A(x₁, y₁) and B(x₂, y₂) in the coordinate system is calculated by the formula:
d = sqrt((x₂ – x₁)² + (y₂ – y₁)² )
The equation of a line in the plane is given in the form y = mx + b, where 'm' is the slope, determining the inclination of the line, and 'b' is the y-intercept of the line. If we know two points on the line, the slope can be calculated as m = (y₂ – y₁) / (x₂ – x₁).
Plane geometry is one of the fundamental branches of mathematics that allows for the analysis and study of figures and their properties. Its concepts are crucial for understanding spatial relationships and are the basis for further exploration of geometry in three dimensions and for use in analytical geometry. Understanding basic concepts such as angles, polygons, circles, and lines facilitates easier solving of geometric problems and broader application in mathematical and scientific disciplines.
