Planimetry is easy. Concepts and formulas

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

After reading the material, the reader will understand that planimetry is not difficult at all. The article provides the most important theoretical information and formulas necessary for solving specific problems. Important statements and properties of figures are put on the shelves.

Definition and important facts

Planimetry is a branch of geometry that considers objects on a flat two-dimensional surface. Some suitable examples can be identified: square, circle, rhombus.

Among other things, it is worth highlighting a point and a line. They are the two basic concepts of planimetry.

Everything else is already built on them, for example:

• A segment is a part of a straight line bounded by two points.
• Ray is an object similar to a segment, however, having a border on one side only.
• An angle that consists of two rays coming out of the same point.

Axioms and theorems

Let's take a closer look at the axioms. In planimetry, these are the most important rules by which all science works. Yes, and not only in it. Byby definition, these are statements that do not require proof.

The axioms that will be discussed below are part of the so-called Euclidean geometry.

• There are two dots. A single line can always be drawn through them.
• If a line exists, then there are points that lie on it and points that do not lie on it.

These 2 statements are called the axioms of membership, and the following ones are of order:

• If there are three points on a straight line, then one of them must be between the other two.
• A plane is divided by any straight line into two parts. When the ends of the segment lie on one half, then the whole object belongs to it. Otherwise, the original line and segment have an intersection point.

Axioms of measures:

• Each segment has a non-zero length. If the point breaks it into several parts, then their sum will be equal to the full length of the object.
• Each angle has a certain degree measure, which is not equal to zero. If you split it with a beam, then the initial angle will be equal to the sum of the formed ones.

Parallel:

There is a straight line on the plane. Through any point that does not belong to it, only one straight line can be drawn parallel to the given one

Theorems in planimetry are no longer quite fundamental statements. They are usually accepted as fact, but each of them has a proof built on the basic concepts mentioned above. Besides, there are a lot of them. It will be quite difficult to disassemble everything, but the material presented will contain someof them.

The following two are worth checking out early:

• The sum of adjacent angles is 180 degrees.
• Vertical angles have the same value.

These two theorems can be useful in solving geometric problems related to n-gons. They are quite simple and intuitive. Worth remembering them.

Triangles

Triangle is a geometric figure consisting of three successively connected segments. They are classified according to several criteria.

On the sides (ratios emerge from the names):

• Equilateral.
• Isosceles - two sides and opposite angles are respectively equal.
• Versatile.

At the corners:

• acute-angled;
• rectangular;
• obtuse.

Two corners will always be sharp regardless of the situation, 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 using the formula: S=½ ⋅ h ⋅ a, where a is the side, h is the height drawn to it.
• You can always inscribe a circle in a triangle or describe it around it.

One of the basic formulas of planimetry is the Pythagorean theorem. It works exclusively for a right triangle and sounds like this: a squarethe hypotenuse is equal to the sum of the squares of the legs: AB² =AC² + BC².

The hypotenuse is the side opposite the 90° angle, and the legs are the adjacent side.

Quadagons

There is a lot of information on this subject. Below are just the most important ones.

Some varieties:

1. Parallelogram - opposite sides are equal and parallel in pairs.
2. Rhombus is a parallelogram whose sides are the same length.
3. Rectangle - parallelogram with four right angles
4. A square is both a rhombus and a rectangle.
5. Trapezium - only two opposite sides are parallel.

Properties:

• The sum of interior angles is 360 degrees.
• The area can always be calculated using the formula: S=√(p-a)(p-b)(p-c)(p-d), where p is half the perimeter, a, b, c, d are the sides of the figure.
• If a circle can be described around a quadrilateral, then I call it convex, if not - non-convex.