A triangle is a geometric figure thatconsists of three points, in turn, they are called vertices, while they are connected to each other sequentially by segments. Such segments are called sides of a triangle. There are several types of triangles, namely:
1. According to the angle:
- obtuse (when one of the angles has a degree measure above ninety degrees);
- rectangular (when one of the corners is ninety degrees);
- acute angle (when all angles have a degree measure less than ninety degrees).
2. By the number of equal parties:
- versatile (all parties differ in size);
- isosceles (the two sides are equal to each other);
- equilateral (all sides have the same length).
Worth noting the fact that the sum of the degree measuresangles in a triangle is always equal to 180 degrees, regardless of the type of the figure itself. So, in an isosceles triangle, the angles that underlie are always equal. And in an equilateral triangle, each angle has exactly sixty degrees. In a right-angled triangle, to find an angle, it is enough to subtract the known angle from ninety degrees. Then all degree measures will be known.
Knowledge of the degree measure of the angle will always give the answer toThe question is how to find the side of a triangle. Consider all the examples of a right triangle, since it is more versatile. In addition, equilateral and isosceles triangles can be easily represented as two rectangular, but more on that later.
The degree measure itself is not enough. It is necessary only in order to be able to calculate the trigonometric relations, namely:
Sin - отношение прилегающего катета к гипотенузе, Cos is the ratio of the opposite leg to the hypotenuse, Tg is the ratio of the adjacent leg to the opposite one, Ctg is the ratio of the opposite leg to the adjacent one.
So, how to find the side of a rectangulartriangle? Knowing the relationship, you can use the sine theorem, which says the following: one side refers to the sine of the angle just as the other side relates to the sine of another angle, and the third side has the same ratio of side and sine of the angle as the previous two.
As can be seen from the theorem, one knowledge of the sines is notenough You need to know the measure of the length of at least one side. Then, how to find the side of the triangle will not cause much difficulty. Or another option is possible. To find one of the legs of the triangle, it is necessary to multiply the hypotenuse either by the sine of the adjacent angle, or by the cosine of the opposite. The value of the party will not change.
In addition, you can use the well-knownthe Pythagorean theorem, which in turn says: the square of the hypotenuse is equal to the sum of the squares of the legs. Here, knowing the two measures of the parties, you can easily determine the value of the third.
There is another theorem on how to findside of the triangle. The cosine theorem: a measure of the length of a side is equal to the square root of the sum of the squares of the other two sides without the double product of these sides, which in turn multiply by the cosine of the angle between them.
And how to find the side of an isosceles triangle? Here, the same principles and theorems have the right to exist as for rectangular ones, but there are several nuances.
First you need to lower the height on the basetriangle. Thus, we obtain two identical rectangular triangles, to which we will apply the previously studied possibilities. How to find the side of a triangle? We will receive both the hypotenuse and two legs. If we have found the hypotenuse, then we already know two sides of the triangle. If we found a leg that is not tall, then multiplying it by two, we get the value of the third party.
Often there are tasks when none of the parties is set. In this case, it is worthwhile to introduce some unknown X, and continue the search for all parties, not paying attention to the replacement of this kind.
