After reading the material, the reader will understand that planimetry is not at all difficult. The article provides the most important theoretical information and formulas necessary for solving specific problems. On the shelves are laid out important statements and properties of figures.
Definition and important facts
Planimetry is a section of geometry that examines objects on a flat two-dimensional surface. You can highlight some suitable examples: square, circle, rhombus.
Among other things, it is worth highlighting a point and a straight line. They are two basic concepts of planimetry.
Everything else is already built on them, for example:
- A line is a part of a straight line bounded by two points.
- A ray is an object similar to a segment, however, having a border on only one side.
- An angle that consists of two rays emanating from one point.
Axioms and Theorems
We will deal with axioms in more detail. In planimetry, these are the most important rules by which all science works. And not only in it. By definition, we are talking about statements that do not require evidence.
The axioms that will be considered below are included in the so-called Euclidean geometry.
- There are two points. Through them you can always draw a single line.
- If there is a line, that is, points that lie on it, and points that do not lie on it.
These 2 statements are called axioms of belonging, and the following - of order:
- If there are three points on a straight line, then one of them is sure to be between the other two.
- A plane is divided into any line into two parts. When the ends of the segment lie on one half, then the whole object also belongs to it. Otherwise, the original line and the segment have an intersection point.
Axioms of measures:
- Each segment has a length other than zero. If the point breaks it into several parts, then their sum will be equal to the total length of the object.
- Each angle has a certain degree measure, which is not equal to zero. If you break it with a beam, then the initial angle will be equal to the sum of the formed.
Parallelism:
- There is a straight line on the plane. Through any point that does not belong to it, you can draw only one straight line parallel to this one.
Planimetry theorems are not quite fundamental statements anymore. Usually they are accepted as a fact, but each of them has a proof built on the basic concepts mentioned above. In addition, there are a lot of them. It will be quite difficult to disassemble everything, but some of them will be present in the presented material.
The following two are worth reading early:
- The sum of adjacent angles is 180 degrees.
- Vertical angles are the same size.
These two theorems can be useful in solving geometric problems related to n-gons. They are quite simple and intuitive. It is worth remembering them.
Triangles
A triangle is a geometric shape consisting of three segments connected in series. They are classified according to several criteria.
On the sides (ratios come up from the names):
- Equilateral.
- Isosceles - two sides and angles opposite to them are respectively equal.
- Versatile.
In the corners:
- acute-angled;
- rectangular;
- obtuse.
Two angles, regardless of the situation, will always be sharp, and the third is determined by the first part of the word. That is, a right triangle has one of the angles equal to 90 degrees.
Properties:
- The larger the angle, the larger the opposite side.
- The sum of all angles is 180 degrees.
- The area can be calculated by the formula: S = ½ ⋅ h ⋅ a, where a is the side, h is the height drawn to it.
- You can always enter a circle in a triangle or describe it around it.
Pythagoras theorem speaks of one of the basic formulas of planimetry. It works exclusively for a right-angled triangle and sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs: AB 2 = AC 2 + BC 2 .
By hypotenuse, we mean the side opposite the angle of 90 °, and by the legs, the adjacent.
Quadrangles
Information on this topic is extremely large. Below is just the most important.
Some varieties:
- Parallelogram - opposite sides are equal and pairwise parallel.
- A rhombus is a parallelogram whose sides have the same length.
- Rectangle - a parallelogram with four right angles
- A square is both a rhombus and a rectangle.
- Trapezoid - only two opposite sides are parallel.
Properties:
- The sum of the internal angles is 360 degrees.
- The area can always be calculated by the formula: S = √ (pa) (pb) (pc) (pd), where p is half the perimeter, a, b, c, d are the sides of the figure.
- If a circle can be described around a quadrangle, then it is called convex, if not, it is non-convex.