Definition of a quadrilateral. Complete Lessons - Knowledge Hypermarket

One of the most interesting topics in geometry from the school course is "Quadangles" (grade 8). What types of such figures exist, what special properties do they have? What is unique about quadrilaterals with ninety-degree corners? Let's look into all this.

What geometric figure is called a quadrilateral

Polygons, which consist of four sides and, accordingly, of four vertices (corners), are called quadrilaterals in Euclidean geometry.

The history of the name of this type of figures is interesting. In the Russian language, the noun "quadrangular" is formed from the phrase "four corners" (in the same way as "triangle" - three corners, "pentagon" - five corners, etc.).

However, in Latin (through which many geometric terms came to most languages ​​of the world), it is called quadrilateral. This word is formed from the numeral quadri (four) and the noun latus (side). So we can conclude that among the ancients this polygon was referred to only as "four-sided".

By the way, such a name (with an emphasis on the presence of four sides rather than corners in figures of this type) has been preserved in some modern languages. For example, in English - quadrilateral and in French - quadrilatère.

At the same time, in most Slavic languages, the considered type of figures is still identified by the number of angles, and not sides. For example, in Slovak (štvoruholník), in Bulgarian (“chetirigalnik”), in Belarusian (“chatyrokhkutnik”), in Ukrainian (“chotirikutnik”), in Czech (čtyřúhelník), but in Polish the quadrilateral is called by the number of sides - czworoboczny.

What types of quadrangles are studied in the school curriculum

In modern geometry, there are 4 types of polygons with four sides.

However, due to the too complex properties of some of them, in geometry lessons, schoolchildren are introduced to only two types.

• Parallelogram. The opposite sides of such a quadrilateral are pairwise parallel to each other and, accordingly, are also equal in pairs.
• Trapeze (trapezium or trapezoid). This quadrilateral consists of two opposite sides parallel to each other. However, the other pair of sides does not have this feature.

Types of quadrilaterals not studied in the school geometry course

In addition to the above, there are two more types of quadrilaterals that schoolchildren are not introduced to in geometry lessons, because of their particular complexity.

• Deltoid (kite)- a figure in which each of two pairs of adjacent sides is equal in length to each other. Such a quadrilateral got its name due to the fact that in appearance it quite strongly resembles the letter of the Greek alphabet - “delta”.
• Antiparallelogram- this figure is as complex as its name. In it, two opposite sides are equal, but at the same time they are not parallel to each other. In addition, the long opposite sides of this quadrilateral intersect each other, as do the extensions of the other two, shorter sides.

Types of parallelogram

Having dealt with the main types of quadrangles, it is worth paying attention to its subspecies. So, all parallelograms, in turn, are also divided into four groups.

• Classical parallelogram.
• Rhombus (rhombus)- a quadrangular figure with equal sides. Its diagonals intersect at right angles, dividing the rhombus into four equal right triangles.
• Rectangle. The name speaks for itself. Since it is a quadrilateral with right angles (each of them is equal to ninety degrees). Its opposite sides are not only parallel to each other, but also equal.
• Square (square). Like a rectangle, it is a quadrilateral with right angles, but it has all sides equal to each other. This figure is close to a rhombus. So it can be argued that a square is a cross between a rhombus and a rectangle.

Rectangle Special Properties

Considering figures in which each of the angles between the sides is equal to ninety degrees, it is worth dwelling more closely on the rectangle. So, what special features does it have that distinguish it from other parallelograms?

To assert that the parallelogram under consideration is a rectangle, its diagonals must be equal to each other, and each of the angles must be right. In addition, the square of its diagonals must correspond to the sum of the squares of two adjacent sides of this figure. In other words, the classical rectangle consists of two right-angled triangles, and in them, as is known, the diagonal of the quadrilateral under consideration acts as the hypotenuse.

The last of the listed signs of this figure is also its special property. Besides this, there are others. For example, the fact that all sides of the studied quadrilateral with right angles are at the same time its heights.

In addition, if a circle is drawn around any rectangle, its diameter will be equal to the diagonal of the inscribed figure.

Among other properties of this quadrilateral, that it is flat and does not exist in non-Euclidean geometry. This is due to the fact that in such a system there are no quadrangular figures, the sum of the angles of which is equal to three hundred and sixty degrees.

Square and its features

Having dealt with the signs and properties of a rectangle, it is worth paying attention to the second quadrilateral known to science with right angles (this is a square).

Being in fact the same rectangle, but with equal sides, this figure has all its properties. But unlike it, the square is present in non-Euclidean geometry.

In addition, this figure has other distinctive features of its own. For example, the fact that the diagonals of a square are not just equal to each other, but also intersect at a right angle. Thus, like a rhombus, a square consists of four right-angled triangles, into which it is divided by diagonals.

In addition, this figure is the most symmetrical among all quadrilaterals.

What is the sum of the angles of a quadrilateral

Considering the features of Euclidean geometry quadrangles, it is worth paying attention to their angles.

So, in each of the above figures, regardless of whether it has right angles or not, their total sum is always the same - three hundred and sixty degrees. This is a unique distinguishing feature of this type of figure.

Perimeter of quadrilaterals

Having figured out what the sum of the angles of a quadrilateral is and other special properties of figures of this type, it is worth knowing what formulas are best used to calculate their perimeter and area.

To determine the perimeter of any quadrilateral, you just need to add together the length of all its sides.

For example, in the KLMN figure, its perimeter can be calculated using the formula: P \u003d KL + LM + MN + KN. If you substitute the numbers here, you get: 6 + 8 + 6 + 8 = 28 (cm).

In the case when the figure in question is a rhombus or a square, to find the perimeter, you can simplify the formula by simply multiplying the length of one of its sides by four: P \u003d KL x 4. For example: 6 x 4 \u003d 24 (cm).

Area quadrilateral formulas

Having figured out how to find the perimeter of any figure with four corners and sides, it is worth considering the most popular and simple ways to find its area.

Other properties of quadrilaterals: inscribed and circumscribed circles

Having considered the features and properties of a quadrilateral as a figure of Euclidean geometry, it is worth paying attention to the ability to describe around or inscribe circles inside it:

• If the sums of the opposite angles of a figure are one hundred and eighty degrees each and are pairwise equal to each other, then a circle can be freely described around such a quadrilateral.
• According to Ptolemy's theorem, if a circle is circumscribed outside a polygon with four sides, then the product of its diagonals is equal to the sum of the products of the opposite sides of the given figure. Thus, the formula will look like this: KM x LN \u003d KL x MN + LM x KN.
• If you construct a quadrilateral in which the sums of opposite sides are equal to each other, then a circle can be inscribed in it.

Having figured out what a quadrilateral is, what types of it exist, which of them have only right angles between the sides and what properties they have, it is worth remembering all this material. In particular, the formulas for finding the perimeter and area of ​​\u200b\u200bthe considered polygons. After all, figures of this form are one of the most common, and this knowledge can be useful for calculations in real life.

1 . The sum of the diagonals of a convex quadrilateral is greater than the sum of its two opposite sides.

2 . If the segments connecting the midpoints of opposite sides quadrilateral

a) are equal, then the diagonals of the quadrilateral are perpendicular;

b) are perpendicular, then the diagonals of the quadrilateral are equal.

3 . The bisectors of the angles at the lateral side of the trapezium intersect at its midline.

4 . The sides of the parallelogram are equal and . Then the quadrilateral formed by the intersections of the bisectors of the angles of the parallelogram is a rectangle whose diagonals are equal.

5 . If the sum of the angles at one of the bases of the trapezoid is 90°, then the segment connecting the midpoints of the bases of the trapezoid is equal to their half-difference.

6 . On the sides AB and AD parallelogram ABCD points are taken M and N so that straight MS and NC Divide the parallelogram into three equal parts. Find MN, if BD=d.

7 . A segment of a straight line parallel to the bases of a trapezoid, enclosed inside the trapezoid, is divided by its diagonals into three parts. Then the segments adjacent to the sides are equal to each other.

8 . Through the point of intersection of the diagonals of the trapezoid with the bases and a straight line is drawn, parallel to the bases. The segment of this line, enclosed between the sides of the trapezoid, is equal to.

9 . A trapezoid is divided by a line parallel to its bases equal to and , into two equal trapezoids. Then the segment of this straight line, enclosed between the sides, is equal to .

10 . If one of the following conditions is met, then four points A, B, C and D lie on the same circle.

a) CAD=CBD= 90°.

b) points BUT and AT lie on one side of a straight line CD and angle CAD equal to the angle CBD

c) straight AC and BD intersect at a point O and O A OS=OV OD.

11 . A line connecting a point R intersections of the diagonals of a quadrilateral ABCD with dot Q line intersections AB and CD, divides the side AD in half. Then she bisects and a side Sun.

12 . Each side of a convex quadrilateral is divided into three equal parts. Corresponding division points on opposite sides are connected by segments. Then these segments divide each other into three equal parts.

13 . Two straight lines divide each of the two opposite sides of a convex quadrilateral into three equal parts. Then between these lines lies one third of the area of ​​the quadrilateral.

14 . If a circle can be inscribed in a quadrilateral, then the segment connecting the points at which the inscribed circle touches opposite sides of the quadrilateral passes through the intersection point of the diagonals.

15 . If the sums of opposite sides of a quadrilateral are equal, then a circle can be inscribed in such a quadrilateral.

16. Properties of an inscribed quadrilateral with mutually perpendicular diagonals. Quadrilateral ABCD inscribed in a circle of radius R. Its diagonals AC and BD are mutually perpendicular and intersect at a point R. Then

a) the median of a triangle ARV perpendicular to the side CD;

b) broken line AOC divides the quadrilateral ABCD into two equal figures;

in) AB 2 +CD 2=4R 2 ;

G) AP 2 + BP 2 + SR 2 + DP 2 = 4R 2 and AB 2 + BC 2 + CD 2 + AD 2 = 8R 2;

e) the distance from the center of the circle to the side of the quadrilateral is half the opposite side.

f) if the perpendiculars dropped to the side AD from the peaks AT and FROM, cross diagonals AC and BD at points E and F, then BCFE- rhombus;

g) a quadrilateral whose vertices are projections of a point R on the side of the quadrilateral ABCD,- both inscribed and described;

h) a quadrilateral formed by tangents to the circumscribed circle of the quadrilateral ABCD, drawn at its vertices can be inscribed in a circle.

17 . If a a, b, c, d- successive sides of a quadrilateral, S- its area, then, and equality takes place only for an inscribed quadrilateral, the diagonals of which are mutually perpendicular.

18 . Brahmagupta formula. If the sides of the inscribed quadrilateral are equal a, b, c and d, then its area S can be calculated by the formula,

where is the semiperimeter of the quadrilateral.

19 . If a quadrilateral with sides a, b, c, d can be inscribed and a circle can be circumscribed around it, then its area is equal to .

20 . Point P is located inside the square ABCD, and the angle PAB equal to the angle RVA and is equal to 15°. Then the triangle DPC- equilateral.

21 . If for an inscribed quadrilateral ABCD equality CD=AD+BC, then the bisectors of its angles BUT and AT intersect on the side CD.

22 . Continuations of opposite sides AB and CD inscribed quadrilateral ABCD intersect at a point M, and the parties AD and sun- at the point N. Then

a) angle bisectors AMD and DNC mutually perpendicular;

b) straight MQ and NQ intersect the sides of the quadrilateral at the vertices of the rhombus;

c) point of intersection Q of these bisectors lies on the segment connecting the midpoints of the diagonals of the quadrilateral ABCD.

23 . Ptolemy's theorem. The sum of the products of two pairs of opposite sides of an inscribed quadrilateral is equal to the product of its diagonals.

24 . Newton's theorem. In any circumscribed quadrilateral, the midpoints of the diagonals and the center of the inscribed circle lie on the same straight line.

25 . Monge's theorem. Lines drawn through the midpoints of the sides of an inscribed quadrilateral perpendicular to opposite sides intersect at one point.

27 . Four circles, built on the sides of a convex quadrilateral as diameters, cover the entire quadrilateral.

29 . Two opposite corners of a convex quadrilateral are obtuse. Then the diagonal connecting the vertices of these angles is less than the other diagonal.

30. The centers of squares built on the sides of a parallelogram outside it form a square themselves.

And again the question is: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has and (remember our sign 2).

And again, since a rhombus is a parallelogram, then it must have all the properties of a parallelogram. This means that a rhombus has opposite angles equal, opposite sides are parallel, and the diagonals are bisected by the point of intersection.

Rhombus Properties

Look at the picture:

As in the case of a rectangle, these properties are distinctive, that is, for each of these properties, we can conclude that we have not just a parallelogram, but a rhombus.

Signs of a rhombus

And pay attention again: there should be not just a quadrangle with perpendicular diagonals, but a parallelogram. Make sure:

No, of course not, although its diagonals and are perpendicular, and the diagonal is the bisector of angles u. But ... the diagonals do not divide, the intersection point in half, therefore - NOT a parallelogram, and therefore NOT a rhombus.

That is, a square is a rectangle and a rhombus at the same time. Let's see what comes out of this.

Is it clear why? - rhombus - the bisector of angle A, which is equal to. So it divides (and also) into two angles along.

Well, it's quite clear: the rectangle's diagonals are equal; rhombus diagonals are perpendicular, and in general - parallelogram diagonals are divided by the point of intersection in half.

AVERAGE LEVEL

Properties of quadrilaterals. Parallelogram

Parallelogram Properties

Attention! The words " parallelogram properties» means that if you have a task there is parallelogram, then all of the following can be used.

Theorem on the properties of a parallelogram.

In any parallelogram:

Let's see why this is true, in other words WE WILL PROVE theorem.

So why is 1) true?

Since it is a parallelogram, then:

• like lying crosswise
• as lying across.

Hence, (on the II basis: and - general.)

Well, once, then - that's it! - proved.

But by the way! We also proved 2)!

Why? But after all (look at the picture), that is, namely, because.

Only 3 left).

To do this, you still have to draw a second diagonal.

And now we see that - according to the II sign (the angle and the side "between" them).

Properties proven! Let's move on to the signs.

Parallelogram features

Recall that the sign of a parallelogram answers the question "how to find out?" That the figure is a parallelogram.

In icons it's like this:

Why? It would be nice to understand why - that's enough. But look:

Well, we figured out why sign 1 is true.

Well, that's even easier! Let's draw a diagonal again.

Which means:

And is also easy. But… different!

Means, . Wow! But also - internal one-sided at a secant!

Therefore the fact that means that.

And if you look from the other side, then they are internal one-sided at a secant! And therefore.

See how great it is?!

And again simply:

Exactly the same, and.

Pay attention: if you found at least one sign of a parallelogram in your problem, then you have exactly parallelogram and you can use everyone properties of a parallelogram.

For complete clarity, look at the diagram:

Properties of quadrilaterals. Rectangle.

Rectangle properties:

Point 1) is quite obvious - after all, sign 3 () is simply fulfilled

And point 2) - very important. So let's prove that

So, on two legs (and - general).

Well, since the triangles are equal, then their hypotenuses are also equal.

Proved that!

And imagine, the equality of the diagonals is a distinctive property of a rectangle among all parallelograms. That is, the following statement is true

Let's see why?

So, (meaning the angles of the parallelogram). But once again, remember that - a parallelogram, and therefore.

Means, . And, of course, it follows from this that each of them After all, in the amount they should give!

Here we have proved that if parallelogram suddenly (!) will be equal diagonals, then this exactly a rectangle.

But! Pay attention! This is about parallelograms! Not any a quadrilateral with equal diagonals is a rectangle, and only parallelogram!

Properties of quadrilaterals. Rhombus

And again the question is: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has and (Remember our sign 2).

And again, since a rhombus is a parallelogram, it must have all the properties of a parallelogram. This means that a rhombus has opposite angles equal, opposite sides are parallel, and the diagonals are bisected by the point of intersection.

But there are also special properties. We formulate.

Rhombus Properties

Why? Well, since a rhombus is a parallelogram, then its diagonals are divided in half.

Why? Yes, that's why!

In other words, the diagonals and turned out to be the bisectors of the corners of the rhombus.

As in the case of a rectangle, these properties are distinctive, each of them is also a sign of a rhombus.

Rhombus signs.

Why is that? And look

Hence, and both these triangles are isosceles.

To be a rhombus, a quadrilateral must first "become" a parallelogram, and then already demonstrate feature 1 or feature 2.

Properties of quadrilaterals. Square

That is, a square is a rectangle and a rhombus at the same time. Let's see what comes out of this.

Is it clear why? Square - rhombus - the bisector of the angle, which is equal to. So it divides (and also) into two angles along.

Well, it's quite clear: the rectangle's diagonals are equal; rhombus diagonals are perpendicular, and in general - parallelogram diagonals are divided by the point of intersection in half.

Why? Well, just apply the Pythagorean Theorem to.

SUMMARY AND BASIC FORMULA

Parallelogram properties:

1. Opposite sides are equal: , .
2. Opposite angles are: , .
3. The angles at one side add up to: , .
4. The diagonals are divided by the intersection point in half: .

Rectangle properties:

1. The diagonals of a rectangle are: .
2. Rectangle is a parallelogram (all properties of a parallelogram are fulfilled for a rectangle).

Rhombus properties:

1. The diagonals of the rhombus are perpendicular: .
2. The diagonals of a rhombus are the bisectors of its angles: ; ; ; .
3. A rhombus is a parallelogram (all properties of a parallelogram are fulfilled for a rhombus).

Square properties:

A square is a rhombus and a rectangle at the same time, therefore, for a square, all the properties of a rectangle and a rhombus are fulfilled. As well as.

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 quadrangle 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 one side of any of its sides.

In contact with

As you can see, the definition is quite easy to remember.

Basic properties and types

Almost all figures known to us, consisting of four corners and sides, can be attributed to convex quadrilaterals. The following can be distinguished:

1. parallelogram;
2. square;
3. rectangle;
4. trapezoid;
5. rhombus.

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.

Is a parallelogram a convex quadrilateral?

Above is an image of a parallelogram. As can be seen from the figure, parallelogram is also convex. If you look at the figure with respect 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 features of a parallelogram are that its sides are pairwise parallel and equal in the same way as opposite angles are equal to each other.

Now, imagine a square or a rectangle. According to their main properties, they are also parallelograms, that is, all their sides are arranged in pairs in parallel. Only in the case of a rectangle, the length 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, while the lengths of the sides and angles of a parallelogram can be different.

As a result, the sum of all four corners of the quadrilateral must be equal to 360 degrees. The easiest way to determine this is by 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.

Property of the diagonals of a convex quadrilateral

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 the convexity of the quadrilateral.

Other properties and signs of convexity of a quadrilateral

Specifically, according to this term, it is very difficult to name any specific properties and features. It is easier to isolate according to different kinds of quadrilaterals of this type. You can start with a parallelogram. We already know that this is a quadrangular figure, the sides of which are pairwise parallel and equal. At the same time, this also includes the property of the diagonals of a parallelogram to intersect with each other, as well as the sign of the convexity of the figure itself: the parallelogram is always in the same plane and on one side relative to any of its sides.

So, the main features and properties are known:

1. the sum of the angles of a quadrilateral is 360 degrees;
2. the diagonals of the figures intersect at one point.

Rectangle. This figure has all the same properties and features as a parallelogram, but all its angles are equal to 90 degrees. Hence the name, rectangle.

Square, the same parallelogram, but its corners 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.

The trapezoid is a very interesting figure.. This is also a quadrilateral and also convex. In this article, the trapezoid has already been considered using the example of a drawing. It is clear that she is also convex. The main difference, and, accordingly, a sign of a trapezoid is that its sides can be absolutely not equal to each other in length, as well as its angles in value. In this case, the figure always remains on the same plane with respect to any of the straight lines that connect any two of its vertices along the segments forming the figure.

Rhombus is an equally interesting figure. Partly 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 the rhombus are equal, then the diagonals are also divided in half at the intersection.

Deltoids or convex rhomboids (rhombuses) may have different side lengths. But at the same time, both the main properties and features of the rhombus itself and the features and properties of convexity are still preserved. That is, we can observe that the diagonals bisect the corners and intersect at right angles.

Today's task was to consider and understand what convex quadrilaterals are, what they are 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 figures, for example, is equal to the sum of the lengths of all segments forming the figure. The formulas for calculating the perimeter and area of ​​quadrilaterals will be discussed in the following articles.

Types of convex quadrilaterals