A triangle is a geometric figure, which consists of three points, in turn they are called vertices, while they are interconnected sequentially by segments. Such segments are called the sides of the triangle. There are several types of triangles, namely:
1. The magnitude of the angles:
- obtuse (when one of the angles has a degree measure above ninety degrees);
- rectangular (when one of the angles is ninety degrees);
- acute-angled (when all angles have a degree measure less than ninety degrees).
2. By the number of equal parties:
- versatile (all sides differ in size);
- isosceles (two sides are equal to each other);
- equilateral (all sides have the same length).
It is worth noting the fact that the sum of degree measures of angles in a triangle is always equal to 180 degrees, regardless of the type of 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 a known angle from ninety degrees. Then all degree measures will be known.
Knowing the degree measure of the angle will always give an answer to the question of how to find the side of the triangle. Let's consider everything on the examples of a right-angled triangle, since it is more universal. In addition, equilateral and isosceles triangles can easily be represented as two rectangular, but more on that later.
The most degree measure is not enough. It is needed only in order to be able to calculate trigonometric relations, namely:
Sin is the ratio of the adjacent leg to the hypotenuse, Cos is the ratio of the opposite leg to the hypotenuse, Tg is the ratio of the adjacent leg to the opposite, Ctg is the ratio of the opposite leg to the adjacent.
So how to find the side of a right triangle? Knowing the relations, we can use the sine theorem, which states the following: one side refers to the sine of the angle in the same way as the other side refers to the sine of the other angle, and the third side has the same ratio of the side and sine of the angle as the two previous ones.
As can be seen from the theorem, just knowing the sines is not enough. You need to know the measure of the length of at least one other 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-known Pythagorean theorem, which in turn states: the square of the hypotenuse is equal to the sum of the squares of the legs. Here, knowing the two measures of the parties, one can easily determine the meaning of the third.
There is another theorem on how to find the side of a triangle. Cosine theorem: A measure of the length of a side is the square root of the sum of the squares of two other sides without a double product of these sides, which in turn are multiplied 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 a rectangular one, but there are several nuances.
First you need to lower the height to the base of the triangle. Thus, we get two identical right-angled triangles, to which we will apply the previously studied possibilities. How to find the side of a triangle? We get both hypotenuse and two legs. If we find the hypotenuse, then we already know the two sides of the triangle. If we find a leg that is not height, then when we multiply it by two, we get the value of the third side.
Often there are tasks when none of the parties is assigned. In this case, it is worth introducing some unknown X, and continue the search for all parties, not paying attention to a replacement of this kind.