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The obtuse triangle: the length of the sides, the sum of the angles. The obtuse triangle described

Still preschool children know what a triangle looks like. But with what they are, the guys are already beginning to understand the school. One type is the obtuse triangle. Understand what it is, the easiest way, if you see a picture with its image. And in theory this is called the "simplest polygon" with three sides and vertices, one of which is an obtuse angle.

Understand the concepts

In geometry, these types of figures with three sides are distinguished: acute, rectangular and obtuse triangles. The properties of these simplest polygons are the same for all. So, for all listed species such inequality will be observed. The sum of the lengths of any two sides will necessarily be greater than the length of the third party.

But in order to be sure that it is a finished figure, and not a set of individual vertices, it is necessary to verify that the basic condition is met: the sum of the angles of the obtuse triangle is 180 ° . The same is true for other types of figures with three sides. True, in the obtuse triangle, one of the angles will be even greater than 90 ° , and the two remaining ones will necessarily be sharp. In this case, the largest angle will be opposite to the longest side. True, this is not all properties of the obtuse triangle. But knowing only these features, students can solve many problems in geometry.

For every polygon with three vertices, it is also true that, continuing on either side, we get an angle whose size will be equal to the sum of two non-adjacent internal vertices. The perimeter of the obtuse triangle is calculated in the same way as for the other figures. It equals the sum of the lengths of all its sides. To determine the area of the triangle, mathematicians derived various formulas, depending on which data are initially present.

Correct design

One of the most important conditions for solving problems in geometry is the correct figure. Often mathematics teachers say that it will help not only visualize what is given and what is required of you, but 80% closer to the correct answer. That's why it's important to know how to build an obtuse triangle. If you just need a hypothetical figure, then you can draw any polygon with three sides so that one of the angles is greater than 90 degrees .

If certain lengths of sides or degrees of angles are given, then to draw an obtuse triangle is necessary in accordance with them. In doing so, it is necessary to try to accurately depict the angles, calculating them with the help of a protractor, and proportionally to the data in the job conditions to display the sides.

Basic lines

Often, students do not have much to know just how these or other figures should look. They can not be limited only to information about which triangle is obtuse and which is rectangular. The mathematics course stipulates that their knowledge of the main features of the figures should be more complete.

So, each student should understand the definition of the bisector, median, middle perpendicular and height. In addition, he must know and their main properties.

So, bisectrixes divide the angle in half, and the opposite side - into segments that are proportional to the adjacent sides.

The median divides any triangle into two equal areas. At the point at which they intersect, each of them is divided into 2 segments in a 2: 1 ratio, if viewed from the top from which it came out. In this case, a large median is always drawn to its smallest side.

No less attention is paid to the height. It is perpendicular to the opposite side from the corner. The height of the obtuse triangle has its own characteristics. If it is drawn from an acute vertex, then it does not fall on the side of this simplest polygon, but on its continuation.

The middle perpendicular is the segment that emerges from the center of the triangle's face. At the same time, it is located at right angles to it.

Working with circles

At the beginning of the study of geometry, it is enough for children to understand how to draw an obtuse triangle, to learn how to distinguish it from other species and remember its basic properties. But senior students of this knowledge is already scarce. For example, on EGE often there are questions about circumscribed and inscribed. The first of these concerns all three vertices of a triangle, and the second has one common point with all sides.

To build an inscribed or described obtuse triangle is already much more complicated, because for this it is necessary first to find out where the center of the circle and its radius should be. By the way, in this case not only a pencil with a ruler, but also a compass becomes a necessary tool.

The same difficulties arise when building inscribed polygons with three sides. Mathematicians have derived various formulas that make it possible to determine their location as accurately as possible.

Inscribed triangles

As mentioned earlier, if a circle passes through all three vertices, then this is called the circumscribed circle. Its main property is that it is the only one. To find out how the circumscribed circle of the obtuse triangle should be located, it must be remembered that its center is at the intersection of three middle perpendiculars that go to the sides of the figure. If in an acute-angled polygon with three vertices this point will be inside it, then in an obtuse polygon it will be inside it.

Knowing, for example, that one of the sides of an obtuse triangle is equal to its radius, one can find an angle that lies opposite to a known face. Its sine will be equal to the result of dividing the length of the known side by 2R (where R is the radius of the circle). That is, the sin angle will be equal to ½. This means that the angle is 150 ° .

If you need to find the radius of the circumscribed circle of the obtuse triangle, then you will need information about the length of its sides (c, v, b) and its area S. After all, the radius is calculated as follows: (c x v x b): 4 x S. By the way, , What kind of figure you are: a versatile obtuse triangle, isosceles, straight or acute. In any situation, thanks to the above formula, you can find out the area of a given polygon with three sides.

Described triangles

Also, quite often you have to work with inscribed circles. According to one of the formulas, the radius of such a figure, multiplied by ½ perimeter, will be equal to the area of the triangle. However, for its clarification you need to know the sides of the obtuse triangle. After all, in order to determine ½ perimeter, you need to add their lengths and divide by 2.

To understand where the center of the circle inscribed in the obtuse triangle should be, three bisectors must be drawn. These are the lines that divide the angles in half. It is at their intersection and will be the center of the circle. In this case, it will be equidistant from each side.

The radius of such a circle inscribed in an obtuse triangle is equal to the square root of the quotient (pc) x (pv) x (pb): p. In this case, p is the half -perimeter of the triangle, c, v, b are its sides.

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