The triangle is one of the fundamentalgeometric shapes, representing three intersecting line segments. This figure was known to scientists of ancient Egypt, ancient Greece and ancient China, who derived most of the formulas and laws used by scientists, engineers and designers until now.
The main components of the triangle include:
• Vertices - intersection points of segments.
• Sides - intersecting segments of straight lines.
Based on these components, they formulateconcepts such as the perimeter of a triangle, its area, the inscribed and circumscribed circle. It is known from school that the perimeter of a triangle is a numerical expression of the sum of all three of its sides. At the same time, there are a great many known formulas for finding this value, depending on the source data that the researcher has in one or another case.
1. The easiest way to find the perimeter of a triangle is used in the case when the numerical values of all three of its sides (x, y, z) are known, as a result:
P = x + y + z
2The perimeter of an equilateral triangle can be found if we recall that in a given figure, all sides, like all angles, are equal. Knowing the length of this side, the perimeter of an equilateral triangle can be determined by the formula:
P = 3x
3In an isosceles triangle, as opposed to an equilateral triangle, only two sides have the same numerical value, so in this case, in general, the perimeter will be as follows:
P = 2x + y
four.The following methods are necessary in cases where the numerical values are not all sides. For example, if the study has data on two sides, and also the angle between them is known, then the perimeter of the triangle can be found using the definition of a third party and a known angle. In this case, this third party will be found by the formula:
z = 2x + 2y-2xycosβ
Based on this, the perimeter of the triangle will be equal to:
P = x + y + 2x + (2y-2xycos β)
five.In the case when the length of not more than one side of the triangle is initially given and the numerical values of the two angles adjacent to it are known, the perimeter of the triangle can be calculated based on the sine theorem:
P = x + sinβ x / (sin (180 ° -β)) + sinγ x / (sin (180 ° -γ))
6. There are cases when known parameters of a circle inscribed into it are used to find the perimeter of a triangle. This formula is also known to most from the school:
P = 2S / r (S is the area of a circle, whereas r is its radius).
From all of the above, it is clear thatThe perimeter of a triangle can be found in many ways, based on the data that the researcher owns. In addition, there are several special cases of finding this value. Thus, the perimeter is one of the most important quantities and characteristics of a right triangle.
As you know, this triangle is calleda figure whose two sides form a right angle. The perimeter of a right-angled triangle is found through a numerical expression of the sum of both legs and the hypotenuse. In the event that the researcher is aware of only two sides, the rest can be calculated using the famous Pythagorean theorem: z = (x2 + y2), if both legs are known, or x = (z2 - y2), if the hypotenuse and the leg are known.
В том случае, если известна длина гипотенузы и one of the corners adjacent to it, the other two sides are found by the formulas: x = z sinβ, y = z cosβ. In this case, the perimeter of a right triangle will be equal to:
P = z (cosβ + sinβ +1)
Also a special case is the calculationthe perimeter of a regular (or equilateral) triangle, that is, such a figure in which all sides and all angles are equal. Calculating the perimeter of such a triangle on a known side does not constitute a problem, however, some other data is often known to the researcher. So, if the radius of the inscribed circle is known, the perimeter of a regular triangle is found by the formula:
P = 6√3r
And if the radius of the circumcircle is given, the perimeter of a regular triangle will be found as follows:
P = 3√3R
Formulas need to remember to successfully apply in practice.
