In the school curriculum in geometry lessons you have to deal with various types quadrilaterals: rhombuses, parallelograms, rectangles, trapezoids, squares. The very first shapes to study are the rectangle and the square.
So what is a rectangle? Definition for 2nd grade secondary school will look like this: this is a quadrilateral with all four corners right. It is easy to imagine what a rectangle looks like: it is a figure with 4 right angles and sides parallel to each other in pairs.
How can we understand, when solving another geometric problem, which quadrilateral we are dealing with? There are three main signs, by which one can unmistakably determine that we are talking about a rectangle. Let's call them:
It is interesting to know: what is convex, its features and symptoms.
Since a rectangle is a parallelogram (i.e., a quadrilateral with pairs of parallel opposite sides), then all its properties and characteristics will be fulfilled for it.
In a rectangle opposite sides are equal and mutually parallel. The longer side is usually called the length (denoted by a), the shorter side is called the width (denoted by b). In the rectangle in the image, the lengths are the sides AB and CD, and the widths are AC and B. D. They are also perpendicular to the bases (i.e., they are the heights).
To find the sides, you can use the formulas below. They accepted symbols: a - the length of the rectangle, b - its width, d - the diagonal (a segment connecting the vertices of two angles lying opposite each other), S - the area of the figure, P - the perimeter, α - the angle between the diagonal and the length, β - sharp corner, which is formed by both diagonals. Methods for finding side lengths:
The perimeter of a quadrilateral is called the sum of the lengths of all its sides. To calculate the perimeter, the following formulas can be used:
Area is the space enclosed by a perimeter. Three main ways to calculate area:
Problems in a school mathematics course often require a good command of properties of the diagonals of a rectangle. We list the main ones:
The following formulas are used to calculate the length of the diagonal:
A square is a special case of a rhombus, parallelogram or rectangle. Its difference from these figures is that all its angles are right and all four sides are equal. Square is regular quadrilateral.
A quadrilateral is called a square in the following cases:
The properties of a square include all the previously discussed properties related to a rectangle, as well as the following:
Here are the frequently used formulas for 
calculations of perimeter, area and square elements:
Let's look at some questions that you may encounter when studying a mathematics course at school, and solve a few simple problems.
Problem 1. How will the area of a rectangle change if the length of its sides is tripled?
Solution : Let us denote the area of the original figure as S0, and the area of a quadrilateral with triple the length of its sides as S1. Using the formula discussed earlier, we obtain: S0 = ab. Now let’s increase the length and width by 3 times and write: S1= 3 a 3 b = 9 ab. Comparing S0 and S1, it becomes obvious that the second area is 9 times larger than the first.
Question 1. Is a quadrilateral with right angles a square?
Solution : From the definition it follows that a figure with right angles is a square only if the lengths of all its sides are equal. In other cases, the figure is a rectangle.
Problem 2. The diagonals of a rectangle form an angle of 60 degrees. The width of the rectangle is 8. Calculate what the diagonal is.
Solution: Recall that the diagonals are divided in half by the point of intersection. Thus, we are dealing with an isosceles triangle with an apex angle of 60°. Since the triangle is isosceles, the angles at the base will also be the same. By simple calculations we find that each of them is equal to 60°. It follows that the triangle is equilateral. The width we know is the base of the triangle, therefore half of the diagonal is also equal to 8, and the length of the whole diagonal is twice as large and equal to 16.
Question 2. Does a rectangle have all sides equal or not?
Solution : Suffice it to remember that all sides must be equal for a square, which is a special case of a rectangle. In all other cases, a sufficient condition is the presence of at least 3 right angles. Equality of the parties is not a mandatory feature.
Problem 3. The area of the square is known and equal to 289. Find the radii of the inscribed and circumscribed circle.
Solution :
Using the formulas for a square, we will carry out the following calculations:
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Quadrilateral ABCD is a figure that consists of four points A, B, C, D, three each, not lying on the same straight line, and four segments AB, BC, CD and AD connecting these points.
The pictures show quadrilaterals.
Points A, B, C and D are called vertices of a quadrilateral, and segments AB, BC, CD and AD - parties. The vertices A and C, B and D are called opposite vertices. Sides AB and CD, BC and AD are called opposing parties.
There are quadrilaterals convex(in the picture - left) and non-convex(in the picture - right).
Each diagonal convex quadrilateral divides it into two triangles(diagonal AC divides ABCD into two triangles ABC and ACD; diagonal BD into BCD and BAD). U non-convex quadrilateral only one of the diagonals divides it into two triangles(diagonal AC divides ABCD into two triangles ABC and ACD; diagonal BD does not).
Let's consider main types of quadrilaterals, their properties, area formulas:
Parallelogram is a quadrilateral whose opposite sides are parallel in pairs.
Properties:
Signs of a parallelogram:
1. If two sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.
2. If in a quadrilateral the opposite sides are equal in pairs, then this quadrilateral is a parallelogram.
3. If in a quadrilateral the diagonals intersect and are divided in half by the point of intersection, then this quadrilateral is a parallelogram.
Area of a parallelogram:
Trapezoid A quadrilateral is called a quadrilateral in which two sides are parallel and the other two sides are not parallel.
Reasons are called parallel sides, and the other two sides are called sides.
Middle line A trapezoid is a segment connecting the midpoints of its sides.
THEOREM.
middle line the trapezoid is parallel to the bases and equal to their half-sum.
Trapezoid area:
Diamond is called a parallelogram in which all sides are equal.
Properties:
Rhombus area:
Rectangle is called a parallelogram in which all angles are equal.
Properties:
Rectangle sign:
If the diagonals of a parallelogram are equal, then this parallelogram is a rectangle.
Rectangle area:
Square is called a rectangle whose sides are all equal.
Properties:
A square has all the properties of a rectangle and a rhombus (a rectangle is a parallelogram, therefore a square is a parallelogram with all sides equal, i.e. a rhombus).
Square area:
A convex quadrilateral is a figure consisting of four sides connected to each other at the vertices, forming four angles together with the sides, while the quadrilateral itself is always in the same plane relative to the straight line on which one of its sides lies. In other words, the entire figure is on the same side of any of its sides.
As you can see, the definition is quite easy to remember.
Almost all known figures consisting of four corners and sides can be classified as convex quadrilaterals. The following can be distinguished:
All these figures are united not only by the fact that they are quadrangular, but also by the fact that they are also convex. Just look at the diagram:
The figure shows a convex trapezoid. Here you can see that the trapezoid is on the same plane or on one side of the segment. If you carry out similar actions, you can find out that in the case of all other sides the trapezoid is convex.
Above is a picture of a parallelogram. As can be seen from the figure, parallelogram is also convex. If you look at the figure relative to the lines on which the segments AB, BC, CD and AD lie, it becomes clear that it is always on the same plane from these lines. The main characteristics of a parallelogram are that its sides are pairwise parallel and equal, just as opposite angles are equal to each other.
Now, imagine a square or rectangle. According to their basic properties, they are also parallelograms, that is, all their sides are located in parallel pairs. Only in the case of a rectangle, the lengths of the sides can be different, and the angles are right (equal to 90 degrees), a square is a rectangle in which all sides are equal and the angles are also right, and in a parallelogram, the lengths of the sides and the angles can be different.
As a result, the sum of all four angles of a quadrilateral should be equal to 360 degrees. The easiest way to determine this is by looking at a rectangle: all four corners of the rectangle are right, that is, equal to 90 degrees. The sum of these 90 degree angles gives 360 degrees, in other words, if you add 90 degrees 4 times, you get the desired result.
The diagonals of a convex quadrilateral intersect. Indeed, this phenomenon can be observed visually, just look at the figure:
The figure on the left shows a non-convex quadrilateral or quadrilateral. As you wish. As you can see, the diagonals do not intersect, at least not all of them. On the right is a convex quadrilateral. Here the property of diagonals to intersect is already observed. The same property can be considered a sign of convexity of a quadrilateral.
It is very difficult to name any specific properties and characteristics using this term. It's easier to separate by various types quadrilaterals of this type. You can start with a parallelogram. We already know that this is a quadrangular figure, the sides of which are parallel and equal in pairs. At the same time, this also includes the property of the diagonals of a parallelogram to intersect each other, as well as the very sign of convexity of the figure: the parallelogram is always in the same plane and on the same side relative to any of its sides.
So, the main features and properties are known:
Rectangle. This figure has all the same properties and characteristics as a parallelogram, but at the same time all its angles are equal to 90 degrees. Hence the name - rectangle.
Square, the same parallelogram, but its angles are right like a rectangle. Because of this, a square is rarely called a rectangle. But the main distinguishing feature of a square, in addition to those already listed above, is that all four of its sides are equal.
Trapezoid is a very interesting figure. This is also a quadrilateral and also convex. In this article, the trapezoid has already been examined using the example of a drawing. It is clear that it is also convex. The main difference, and therefore a sign of a trapezoid, is that its sides may be absolutely unequal to each other in length, as well as its angles in value. In this case, the figure always remains on the same plane relative to any of the lines that connects any two of its vertices along the segments forming the figure.
A rhombus is an equally interesting figure. In part, a rhombus can be considered a square. A sign of a rhombus is the fact that its diagonals not only intersect, but also divide the corners of the rhombus in half, and the diagonals themselves intersect at right angles, that is, they are perpendicular. If the lengths of the sides of a rhombus are equal, then the diagonals are also divided in half when they intersect.
Deltoids or convex rhomboids (rhombuses) may have different side lengths. But at the same time, both the basic properties and characteristics of the rhombus itself, as well as the characteristics and properties of convexity, are still preserved. That is, we can observe that the diagonals bisect the angles and intersect at right angles.
Today’s task was to consider and understand what convex quadrilaterals are, what they are like and their main features and properties. Attention! It is worth recalling once again that the sum of the angles of a convex quadrilateral is 360 degrees. The perimeter of the figures, for example, equal to the sum the lengths of all segments forming the figure. Formulas for calculating the perimeter and area of quadrilaterals will be discussed in the following articles.
Today we'll look at geometric figure- quadrangle. From the name of this figure it already becomes clear that this figure has four corners. But we will consider the remaining characteristics and properties of this figure below.
A quadrilateral is a polygon consisting of four points (vertices) and four segments (sides) connecting these points in pairs. The area of a quadrilateral is equal to half the product of its diagonals and the angle between them.
A quadrilateral is a polygon with four vertices, three of which do not lie on a straight line.
Self-intersecting
Non-convex
Convex
Self-intersecting quadrilateral is a quadrilateral in which any of its sides have an intersection point (in blue in the figure).
Non-convex quadrilateral is a quadrilateral in which one of internal corners more than 180 degrees (indicated in orange in the figure).
Sum of angles any quadrilateral that is not self-intersecting is always equal to 360 degrees.
Quadrilaterals can have additional properties, forming special types geometric shapes:
A quadrilateral circumscribed around a circle (a circle inscribed in a quadrilateral).
The main property of the described quadrilateral:
A quadrilateral can be circumscribed around a circle if and only if the sums of the lengths of opposite sides are equal.
Quadrilateral inscribed in a circle (circle circumscribed around a quadrilateral)
The main property of an inscribed quadrilateral:
A quadrilateral can be inscribed in a circle if and only if the sum of the opposite angles is equal to 180 degrees.
Modulus of the difference between any two sides of a quadrilateral does not exceed the sum of its other two sides.
|a - b| ≤ c + d
|a - c| ≤ b + d
|a - d| ≤ b + c
|b - c| ≤ a + d
|b - d| ≤ a + b
|c - d| ≤ a + b
Important. The inequality is true for any combination of sides of a quadrilateral. The figure is provided solely for ease of perception.
In any quadrilateral the sum of the lengths of its three sides is not less than the length of the fourth side.
Important. When solving problems within school curriculum you can use strict inequality (<). Равенство достигается только в случае, если четырехугольник является "вырожденным", то есть три его точки лежат на одной прямой. То есть эта ситуация не попадает под классическое определение четырехугольника.
