How to build an isosceles triangle. Obtuse triangle: length of sides, sum of angles. Obtuse Triangle Circumscribed How to Construct an Acute Triangle

More kids preschool age 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 obtuse triangle. To understand what it is, the easiest way is to see a picture with its image. And in theory, this is what they call the "simplest polygon" with three sides and vertices, one of which is

Understanding concepts

In geometry, there are such types of figures with three sides: acute-angled, right-angled and obtuse-angled triangles. Moreover, the properties of these simplest polygons are the same for all. Yes, for everyone listed species such a disparity will hold. The sum of the lengths of any two sides is necessarily greater than the length of the third side.

But in order to be sure that we are talking it is about a complete figure, and not about a set of individual vertices, that it is necessary to check that the main condition is observed: the sum of the angles of an obtuse triangle is 180 o. 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 o, and the remaining two will necessarily be sharp. In this case, 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 even knowing only these features, students can solve many problems in geometry.

For every polygon with three vertices, it is also true that by continuing any of the sides, we get an angle whose size will be is equal to the sum two internal vertices not adjacent to it. The perimeter of an obtuse triangle is calculated in the same way as for other shapes. It is equal to the sum of the lengths of all its sides. To determine the mathematicians, various formulas were derived, depending on what data was initially present.

Correct style

One of the most important conditions for solving problems in geometry is the correct drawing. Mathematics teachers often say that it will help not only visualize what is given and what is required of you, but also get 80% closer to the correct answer. That is why it is important to know how to construct an obtuse triangle. If you just want 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 values of the lengths of the sides or degrees of angles are given, then it is necessary to draw an obtuse-angled triangle in accordance with them. At the same time, it is necessary to try to depict the angles as accurately as possible, calculating them with the help of a protractor, and display the sides in proportion to the given conditions in the task.

Main lines

Often, it is not enough for schoolchildren to know only how certain figures should look. They cannot limit themselves to information about which triangle is obtuse and which is right-angled. 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 the bisector, median, perpendicular bisector and height. In addition, he must know their basic properties.

So, the 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 areas. At the point at which they intersect, each of them is divided into 2 segments in a ratio of 2: 1, when viewed from the top from which it originated. In this case, the largest median is always drawn to its smallest side.

No less attention is paid to height. This is perpendicular to the opposite side from the corner. The height of an obtuse triangle has its own characteristics. If it is drawn from a sharp vertex, then it falls not on the side of this simplest polygon, but on its extension.

The perpendicular bisector is the line segment that comes out of the center of the face of the triangle. At the same time, it is located at a right angle 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-angled triangle, learn to distinguish it from other types and remember its basic properties. But for high school students this knowledge is not enough. For example, at the exam, there are often questions about circumscribed and inscribed circles. The first of them touches all three vertices of the triangle, and the second has one common point with all sides.

Constructing an inscribed or circumscribed obtuse-angled triangle is already much more difficult, because for this you first need 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 will become a necessary tool.

The same difficulties arise when constructing inscribed polygons with three sides. Mathematicians have developed various formulas that allow you to determine their location as accurately as possible.

Inscribed Triangles

As mentioned earlier, if the 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 an obtuse triangle should be located, it must be remembered that its center is at the intersection of the three median 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 one - outside it.

Knowing, for example, that one of the sides of an obtuse triangle is equal to its radius, one 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 equal to ½. So the angle will be 150 o.

If you need to find the radius of the circumscribed circle of an obtuse-angled 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 like this: (c x v x b): 4 x S. By the way, it doesn’t matter what kind of figure do you have: a versatile obtuse triangle, isosceles, right or acute. In any situation, thanks to the above formula, you can find out the area of a given polygon with three sides.

Circumscribed Triangles

It is also quite common to work with inscribed circles. According to one of the formulas, the radius of such a figure, multiplied by ½ of the perimeter, will equal the area of the triangle. True, to find it out, 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 triangle should be, it is necessary to draw three bisectors. These are the lines that bisect the corners. It is at their intersection that the center of the circle will be located. 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 quotient (p-c) x (p-v) x (p-b) : p. Moreover, p is the half-perimeter of the triangle, c, v, b are its sides.

How to draw a triangle?

Building various triangles- obligatory element school course geometry. For many, this task is intimidating. But in fact, everything is quite simple. The rest of the article describes how to draw any type of triangle using a compass and straightedge.

Triangles are

versatile;
isosceles;
equilateral;
rectangular;
obtuse;
acute-angled;
inscribed in a circle;
circumscribed around a circle.

Construction of an equilateral triangle

An equilateral triangle is a triangle in which all sides are equal. Of all the types of triangles, drawing an equilateral one is the easiest.

Using a ruler, draw one of the sides of a given length.
Measure its length with a compass.
Place the point of the compass at one end of the line and draw a circle.
Move the tip to the other end of the segment and draw a circle.
We have 2 points of intersection of the circles. Connecting any of them with the edges of the segment, we get an equilateral triangle.

Construction of an isosceles triangle

This type of triangles can be built on the base and sides.

An isosceles triangle is one in which two sides are equal. In order to draw an isosceles triangle according to these parameters, you must perform the following steps:

Using a ruler, set aside a segment equal in length to the base. We denote it by the letters AC.
With a compass we measure the required length of the side.
We draw from point A, and then from point C, circles whose radius is equal to the length of the side.
We get two points of intersection. By connecting one of them with points A and C, we get the necessary triangle.

Construction of a right triangle

A triangle with one right angle is called a right triangle. If we are given a leg and a hypotenuse, it will not be difficult to draw a right triangle. It can be built along the leg and hypotenuse.

Construction of an obtuse-angled triangle given an angle and two adjacent sides

If one of the angles of a triangle is obtuse (greater than 90 degrees), it is called an obtuse angle. To draw an obtuse triangle according to the specified parameters, you must do the following:

Using a ruler, set aside a segment equal in length to one of the sides of the triangle. Let's call it A and D.
If an angle has already been drawn in the task, and you need to draw the same one, then on its image set aside two segments, both ends of which lie at the vertex of the angle, and the length is equal to the specified sides. Connect the dots. We have the required triangle.
To transfer it to your drawing, you need to measure the length of the third side.

Construction of an acute triangle

An acute triangle (all angles less than 90 degrees) is built on the same principle.

Draw two circles. The center of one of them lies at point D, and the radius is equal to the length of the third side, while the center of the second is at point A, and the radius is equal to the length of the side specified in the task.
Connect one of the intersection points of the circle with points A and D. The desired triangle is built.

inscribed triangle

In order to draw a triangle in a circle, you need to remember the theorem, which says that the center of the circumscribed circle lies at the intersection of the perpendicular bisectors:

For an obtuse triangle, the center of the circumscribed circle lies outside the triangle, and for a right triangle, it lies in the middle of the hypotenuse.

Draw a circumscribed triangle

The described triangle is a triangle in the center of which a circle is drawn, touching all its sides. The center of the inscribed circle lies at the intersection of the bisectors. To build them you need:

Instruction

Place the needle of the compass at the marked point. Draw a circle with a stylus with a measured radius.

Place a dot anywhere along the circumference of the drawn arc. This will be the second vertex B of the triangle being created.

Place the leg on the second vertex in the same way. Draw another circle so that it intersects with the first.

The third vertex C of the created triangle is located at the intersection point of both drawn arcs. Mark it on the picture.

Having obtained all three vertices, connect them with straight lines using any flat surface (better than a ruler). Triangle ABC is built.

If the circle touches all three sides given triangle, and its center is inside the triangle, then it is called inscribed in the triangle.

You will need

ruler, circle

Instruction

From the vertices of the triangle (the side opposite to the divisible angle), arcs of a circle of arbitrary radius are drawn with a compass until they intersect with each other;

The point of intersection of the arcs along the ruler is connected to the top of the divisible angle;

The same is done with any other angle;

The radius of a circle inscribed in a triangle will be the ratio of the area of the triangle and its semi-perimeter: r=S/p, where S is the area of the triangle, and p=(a+b+c)/2 is the semi-perimeter of the triangle.

The radius of a circle inscribed in a triangle is equidistant from all sides of the triangle.

Sources:

http://www.alleng.ru/d/math/math42.htm

Consider the problem of constructing a triangle, provided that three of its sides or one side and two angles are known.

You will need

- compasses
- ruler
- protractor

Instruction

Let's say there are three sides: a, b and c. Using, it is not difficult with such parties. First, let's choose the longest of these sides, let it be side c, and draw it. Then we set the opening of the compass to the value of the other side, side a, and draw with the compass a circle of radius a centered on one of the ends of side c. Now set the opening of the compass to the value of side b and draw a circle centered on the other end of side c. The radius of this circle is b. We connect the point of intersection of the circles with the centers and get a triangle with the desired sides.

Use a protractor to draw a triangle with a given side and two adjacent angles. Draw a side of the specified length. At the edges of it, set aside the corners with a protractor. At the intersection of the sides of the corners, get the third vertex of the triangle.

Related videos

A triangle is a polygon with three sides. An equilateral or regular triangle is a triangle in which all sides and angles are equal. Consider how you can draw a regular triangle.

You will need

Ruler, circle.

Instruction

Using a compass, draw another circle, the center of which will be at point B, and the radius is equal to the line segment BA.

The circles will intersect at two points. Choose any of them. Name it C. This will be the third vertex of the triangle.

Connect the vertices together. The resulting triangle will be correct. Verify this by measuring its sides with a ruler.

Consider a method for constructing a regular triangle using two rulers. Draw the segment OK, it will be one of the sides of the triangle, and the points O and K will be its vertices.

Without moving the ruler after constructing the OK segment, attach another ruler perpendicular to it. Draw a line m intersecting the segment OK in the middle.

Using a ruler, measure the segment OE, equal to the segment OK so that one of its ends coincides with the point O, and the other is on the line m. Point E will be the third vertex of the triangle.

Finish the construction of the triangle by connecting the points E and K. Check the construction with a ruler.

note

You can make sure that the triangle is correct using a protractor by measuring the angles.

Helpful advice

An equilateral triangle can also be drawn on a sheet in a cage using a single ruler. Instead of another ruler, use perpendicular lines.

Sources:

Classification of triangles. Equilateral triangles
What is a triangle
construction of a right triangle

An inscribed triangle is a triangle all of whose vertices are on the circle. You can build it if you know at least one side and an angle. The circle is called circumscribed, and it will be the only one for this triangle.

You will need

- circle;
- side and angle of a triangle;
- paper;
- compass;
- ruler;
- protractor;
- calculator.

Instruction

From point A, use a protractor to set aside the given angle. Continue the side of the corner to the intersection with the circle and put a point C. Connect the points B and C. You have a triangle ABC. It can be of any type. The center of the circle at an acute triangle is outside it, at an obtuse triangle it is outside, and at a right triangle it is on the hypotenuse. If you are given not an angle, but, for example, three sides of a triangle, calculate one of the angles from the radius and the known side.

Much more often one has to deal with inverse construction when a triangle is given and a circle must be described around it. Calculate its radius. This can be done according to several formulas, depending on what is given to you. The radius can be found, for example, by the side and sine of the opposite angle. In this case, it is equal to the length of the side divided by twice the sine of the opposite angle. That is, R=a/2sinCAB. It can also be expressed through the product of the sides, in this case R=abc/√(a+b+c)(a+b-c)(a+c-b)(b+c-a).

Determine the center of the circle. Divide all sides in half and draw perpendiculars to the middle. The point of their intersection will be the center of the circle. Draw it so that it intersects all the vertices of the corners.

two short sides right triangle, which are commonly called legs, by definition must be perpendicular to each other. This property of the figure greatly facilitates its construction. However, it is not always possible to accurately determine perpendicularity. In such cases, you can calculate the lengths of all sides - they will allow you to build a triangle in the only possible, and therefore correct, way.

You will need

Paper, pencil, ruler, protractor, compass, square.

How to build an isosceles triangle? This is easy to do with a ruler, pencil and notebook cells.

We start building an isosceles triangle from the base. To make the drawing even, the number of cells at the base must be an even number.

We divide the segment - the base of the triangle - in half.

The vertex of the triangle can be chosen at any height from the base, but always exactly above the middle.

How to construct an acute isosceles triangle?

The angles at the base of an isosceles triangle can only be acute. In order for an isosceles triangle to turn out to be acute, the angle at the vertex must also be acute.

To do this, select the top of the triangle higher, away from the base.

The higher the top, the less angle at the top. At the same time, the angles at the base increase accordingly.

How to construct an obtuse isosceles triangle?

As the apex of an isosceles triangle approaches the base, the degree measure of the angle at the apex increases.

So, to build an isosceles obtuse-angled triangle, we choose a vertex lower.

How to construct an isosceles right triangle?

To build an isosceles right triangle, you need to select the vertex at a distance equal to half the base (this is due to the properties of an isosceles right triangle).

For example, if the length of the base is 6 cells, then we place the top of the triangle at a height of 3 cells above the middle of the base. Please note: in this case, each cell at the corners at the base is divided diagonally.

The construction of an isosceles right triangle can be started from the top.

We select the top, from it at a right angle we set aside equal segments up and to the right. These are the sides of the triangle.

Connect them and get an isosceles right triangle.

The construction of an isosceles triangle using a compass and a ruler without divisions will be considered in another topic.