Even preschool children know what a triangle looks like. But with what they are, the guys are already starting to understand at school. One of the types is an obtuse triangle. To understand what it is, the easiest way is 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.
Understanding concepts
In geometry, these types of figures are distinguished with three sides: acute-angled, rectangular and obtuse-angled triangles. Moreover, the properties of these simplest polygons are the same for everyone. So, for all the 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 side.
But in order to be sure that we are talking about a complete figure, and not about a set of individual vertices, it is necessary to verify that the basic condition is met: the sum of the angles of an obtuse-angled triangle is 180
° . The same is true for other types of figures with three sides. True, in an obtuse triangle one of the angles will be even more than 90
° , and the two remaining ones will certainly be sharp. Moreover, it is the largest angle that will be opposite the longest side. True, these are far from all the properties of an obtuse triangle. But knowing only these features, students can solve many problems in geometry.
For each 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 internal vertices that are not adjacent to it. The perimeter of an obtuse triangle is calculated in the same way as for other figures. It equals the sum of the lengths of all its sides. To determine the area of the triangle, mathematicians have deduced various formulas, depending on which data is initially present.
The correct style
One of the most important conditions for solving problems in geometry is the correct pattern. Often, teachers of mathematics say that it will help not only to visualize what is given and what is required of you, but by 80% to come closer to the correct answer. That is why it is 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 corners is more than 90 ° .
If certain values of the lengths of the sides or degrees of angles are given, then an obtuse-angled triangle must be drawn in accordance with them. At the same time, it is necessary to try to depict angles as accurately as possible, calculating them using a protractor, and display the sides in proportion to the conditions in the task.
Main lines
Often, it’s not enough for schoolchildren to know just how certain figures should look. They cannot be limited only to information about which triangle is obtuse and which is rectangular. The course of mathematics provides that their knowledge of the main features of the figures should be more complete.
So, each student should understand the definition of a bisector, median, mid-perpendicular and height. In addition, he must know their basic properties.
So, bisectors 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 in area. At the point at which they intersect, each of them is divided into 2 segments in a 2: 1 ratio, when viewed from the top from which it emerged. Moreover, a large median is always drawn to its smallest side.
No less attention is paid to height. It is perpendicular to the side opposite from the corner. The height of an obtuse triangle has its own characteristics. If it is drawn from a sharp peak, then it falls not on the side of this simplest polygon, but on its continuation.
The mid-perpendicular is the line that extends from the center of the face of the triangle. Moreover, it is located at a right angle to it.
Work with circles
At the beginning of the study of geometry, it is enough for children to understand how to draw an obtuse-angled triangle, learn to distinguish it from other types and remember its basic properties. But high school students this knowledge is not enough. For example, at the USE there are often questions about the circumscribed and inscribed circles. The first of them concerns all three vertices of the triangle, and the second has one common point with all sides.
To build an inscribed or described obtuse-angled triangle is already much more difficult, because for this it is necessary to first 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 will become a necessary tool.
The same difficulties arise when building inscribed polygons with three sides. Mathematicians have derived various formulas that allow us 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 a circumscribed circle. Its main property is that it is the only one. To find out how the described circle of an obtuse-angled triangle should be located, it is necessary to remember that its center is at the intersection of the 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-angled polygon - outside it.
Knowing, for example, that one of the sides of an obtuse-angled triangle is equal to its radius, you can find the angle that lies opposite the 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 of the angle will be ½. So, the angle will be equal to 150 about .
If you need to find the radius of the circumscribed circle of an obtuse-angled triangle, then information about the length of its sides (c, v, b) and its area S is useful. After all, the radius is calculated like this: (c x v x b): 4 x S. By the way, it doesn’t matter what kind of figure you have: a versatile obtuse triangle, isosceles, right-angled or acute-angled. 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 of the perimeter, will be equal to the area of the triangle. True, to clarify it, you need to know the sides of an obtuse triangle. Indeed, in order to determine ½ of the perimeter, it is necessary to add their lengths and divide by 2.
To understand where the center of a circle inscribed in an obtuse-angled triangle should be located, it is necessary to draw three bisectors. These are lines that divide angles in half. It is at their intersection that the center of the circle will be located. Moreover, he will be equidistant from each side.
The radius of such a circle inscribed in an obtuse triangle equals the square root of the quotient (pc) x (pv) x (pb): p. Moreover, p is the semiperimeter of the triangle, c, v, b are its sides.