Theory on the topic triangle preparation for the Unified State Exam. Basic objects and axioms of planimetry

Lesson topic

Trapezoid

Lesson Objectives

Continue to introduce new definitions in geometry;
Consolidate knowledge about already studied geometric shapes;
Introduce the formulation and evidence of the properties of the trapezoid;
Teach the use of the properties of various figures when solving problems and completing assignments;
Continue to develop attention in schoolchildren, logical thinking And math speech;
Cultivate interest in the subject.

Lesson Objectives

Arouse interest in knowledge of geometry;
Continue to train students in solving problems;
Call cognitive interest for math lessons.

Lesson Plan

1. Review the material studied earlier.
2. Introduction to the trapezoid, its properties and characteristics.
3. Solving problems and completing assignments.

Repetition of previously studied material

In the previous lesson, you were introduced to such a figure as a quadrilateral. Let's consolidate the material covered and answer the questions posed:

1. How many angles and sides does a tetragon have?
2. Formulate the definition of a 4-gon?
3. What is the name? opposite sides 4-gon?
4. What types of quadrilaterals do you know? List them and define each of them.
5. Draw an example of a convex and non-convex quadrilateral.

Trapezoid. General properties and definition

A trapezoid is a quadrangular figure in which only one pair of opposite sides is parallel.

IN geometric definition A trapezoid is a tetragon that has two parallel sides, but the other two are not.

The name is like this unusual figure, as “trapezoid” comes from the word “trapezion”, which is translated from Greek language, denotes the word “table”, from which the word “meal” and other related words also come.

In some cases in a trapezoid, a pair of opposite sides are parallel, but its other pair is not parallel. In this case, the trapezoid is called curvilinear.

Trapezoid elements

A trapezoid consists of elements such as a base, lateral lines, midline and its height.

The base of a trapezoid is its parallel sides;
The lateral sides are the other two sides of the trapezoid that are not parallel;
The midline of a trapezoid is the segment that connects the midpoints of its sides;
The height of a trapezoid is the distance between its bases.

Types of trapezoids

Exercise:

1. Formulate the definition of an isosceles trapezoid.
2. Which trapezoid is called rectangular?
3. What does an acute-angled trapezoid mean?
4. Which trapezoid is an obtuse one?

General properties of a trapezoid

Firstly, the midline of the trapezoid is parallel to the base of the figure and is equal to its half-sum;

Secondly, the segment that connects the midpoints of the diagonals of a 4-gonal figure is equal to the half-difference of its bases;

Thirdly, in a trapezoid, parallel lines that intersect the sides of the angle of a given figure are cut off proportional segments from the sides of the corner.

Fourthly, in any type of trapezoid, the sum of the angles that are adjacent to its side is equal to 180°.

Where else is the trapezoid present?

The word "trapezoid" is present not only in geometry, it has a wider application in everyday life.

This unusual word We can meet, while watching sports competitions, gymnasts performing acrobatic exercises on the trapeze. In gymnastics, a trapeze is a sports apparatus that consists of a crossbar suspended on two ropes.

You can also hear this word when working out in the gym or among people who are involved in bodybuilding, since the trapezius is not only a geometric figure or a sports acrobatic apparatus, but also powerful back muscles that are located at the back of the neck.

The picture shows an aerial trapeze, which was invented for circus acrobats by artist Julius Leotard back in the nineteenth century in France. At first, the creator of this act installed his projectile at a low altitude, but in the end it was moved right under the circus dome.

Aerialists in the circus perform tricks of flying from trapeze to trapeze, perform cross flights, and perform somersaults in the air.

In equestrian sports, trapeze is an exercise for stretching or stretching the horse's body, which is very useful and pleasant for the animal. When the horse stands in the trapezoid position, stretching the animal's legs or back muscles works. This nice exercise we can observe during the bow or the so-called “front crunch”, when the horse bends deeply.

Assignment: Give your own examples of where else in everyday life you can hear the words “trapezoid”?

Did you know that for the first time in 1947, the famous French fashion designer Christian Dior held a fashion show in which the silhouette of an a-line skirt was present. And although more than sixty years have passed, this silhouette is still in fashion and does not lose its relevance to this day.

In the wardrobe Queen of England the A-line skirt became an indispensable item and her calling card.

Reminiscent geometric shape A-line skirt of the same name goes perfectly with any blouses, blouses, tops and jackets. The classicism and democratic nature of this popular style allows it to be worn with formal jackets and slightly frivolous tops. It would be appropriate to wear such a skirt both in the office and at a disco.

Problems with trapezoid

To make solving problems with trapezoids easier, it is important to remember a few basic rules:

First, draw two heights: BF and CK.

In one of the cases, as a result you will get a rectangle - ВСФК, from which it is clear that FК = ВС.

AD=AF+FK+KD, hence AD=AF+BC+KD.

In addition, it is immediately obvious that ABF and DCK are right triangles.

Another option is possible when the trapezoid is not quite standard, where

AD=AF+FD=AF+FK–DK=AF+BC–DK.

But the simplest option is if our trapezoid is isosceles. Then solving the problem becomes even easier, because ABF and DCK are right triangles and they are equal. AB=CD, since the trapezoid is isosceles, and BF=CK, as the height of the trapezoid. From the equality of triangles follows the equality of the corresponding sides.

A trapezoid is called convex quadrilateral, in which one pair of opposite sides are parallel to each other, and the other is not.

Based on the definition of a trapezoid and the characteristics of a parallelogram, the parallel sides of a trapezoid cannot be equal to each other. Otherwise, the other pair of sides would also become parallel and equal to each other. In this case, we would be dealing with a parallelogram.

The parallel opposite sides of a trapezoid are called reasons. That is, the trapezoid has two bases. Non-parallel opposite sides of a trapezoid are called sides.

Depending on which lateral sides, which angles they form with the bases, they are distinguished various types trapezoid. Most often, trapezoids are divided into unequal (unilateral), isosceles (equilateral) and rectangular.

U lopsided trapezoids the sides are not equal to each other. Moreover, with a large base, both of them can form only acute angles, or one angle will be obtuse and the other acute. In the first case, the trapezoid is called acute-angled, in the second - obtuse.

U isosceles trapezoids the sides are equal to each other. Moreover, with a large base they can only form acute angles, i.e. All isosceles trapezoids are acute-angled. Therefore, they are not divided into acute-angled and obtuse-angled.

U rectangular trapezoids one side perpendicular to the bases. The second side cannot be perpendicular to them, because in this case we would be dealing with a rectangle. In rectangular trapezoids, the non-perpendicular side always forms an acute angle with the larger base. A perpendicular side is perpendicular to both bases because the bases are parallel.

Trapezoid. Definition, formulas and properties

A trapezoid is a quadrilateral whose pair of opposite sides are parallel.

Note. In this case, the parallelogram is a special case of a trapezoid.

The parallel opposite sides are called the bases of the trapezoid, and the other two are called the lateral sides.

Trapezes are:

- versatile ;

- isosceles;

- rectangular

A - isosceles (isosceles, isosceles) trapezoid
B- rectangular trapezoid
C - scalene trapezoid

A scalene trapezoid has all sides of different lengths and the bases are parallel.

The sides are equal and the bases are parallel.

The bases are parallel, one side is perpendicular to the bases, and the second side is inclined to the bases.

Properties of a trapezoid

• Midline of trapezoid parallel to the bases and equal to their half-sum
• A segment connecting the midpoints of the diagonals, equal to half difference of bases and lies on midline. Its length
• Parallel lines intersecting the sides of any angle of a trapezoid cut off proportional segments from the sides of the angle (see Thales' Theorem)
• Point of intersection of trapezoid diagonals, the intersection point of the extensions of its sides and the middle of its bases lie on the same straight line (see also properties of a quadrilateral)
• Triangles lying on bases trapezoids whose vertices are the intersection point of its diagonals are similar. The ratio of the areas of such triangles is equal to the square of the ratio of the bases of the trapezoid
• Triangles lying on the sides trapezoids whose vertices are the intersection point of its diagonals are equal in area (equal in area)
• Into the trapeze you can inscribe a circle, if the sum of the lengths of the bases of a trapezoid is equal to the sum of the lengths of its sides. The middle line in this case is equal to the sum of the sides divided by 2 (since the middle line of a trapezoid is equal to half the sum of the bases)
• Segment parallel to the bases and passing through the point of intersection of the diagonals, is divided by the latter in half and is equal to twice the product of the bases divided by their sum 2ab / (a ​​+ b) (Burakov’s formula)

Trapezoid angles

A rectangular trapezoid has two right angles, and the other two are acute and obtuse. Other types of trapezoids have: two acute angles and two stupid ones.

Obtuse angles trapezoids belong to the smaller one along the length of the base, and spicy - more basis.

Any trapezoid can be considered like a truncated triangle, whose section line is parallel to the base of the triangle.
Important. Please note that in this way ( additional construction trapezoids to triangles) some problems about trapezoids can be solved and some theorems can be proven.

How to find the sides and diagonals of a trapezoid

Finding the sides and diagonals of a trapezoid is done using the formulas given below:

In these formulas, the notation used is as in the figure.

a - the smaller of the bases of the trapezoid
b - the larger of the bases of the trapezoid
c,d - sides
h 1 h 2 - diagonals

The sum of the squares of the diagonals of a trapezoid is equal to twice the product of the bases of the trapezoid plus the sum of the squares of the lateral sides (Formula 2)

1. The segment connecting the midpoints of the diagonals of a trapezoid is equal to half the difference of the bases
2. Triangles formed by the bases of a trapezoid and the segments of the diagonals up to their point of intersection are similar
3. Triangles formed by segments of the diagonals of a trapezoid, the sides of which lie on the lateral sides of the trapezoid - equal in size (have the same area)
4. If you extend the sides of the trapezoid to the side smaller base, then they intersect at one point with the line connecting the midpoints of the bases
5. A segment connecting the bases of a trapezoid and passing through the point of intersection of the diagonals of the trapezoid is divided by this point in a proportion equal to the ratio of the lengths of the bases of the trapezoid
6. A segment parallel to the bases of the trapezoid and drawn through the point of intersection of the diagonals is divided in half by this point, and its length is equal to 2ab/(a + b), where a and b are the bases of the trapezoid

Properties of a segment connecting the midpoints of the diagonals of a trapezoid

Let's connect the midpoints of the diagonals of the trapezoid ABCD, as a result of which we will have a segment LM.
A segment connecting the midpoints of the diagonals of a trapezoid lies on the midline of the trapezoid.

This segment parallel to the bases of the trapezoid.

The length of the segment connecting the midpoints of the diagonals of a trapezoid is equal to half the difference of its bases.

LM = (AD - BC)/2
or
LM = (a-b)/2

Properties of triangles formed by the diagonals of a trapezoid

Triangles that are formed by the bases of a trapezoid and the point of intersection of the diagonals of the trapezoid - are similar.
Triangles BOC and AOD are similar. Since angles BOC and AOD are vertical, they are equal.
Angles OCB and OAD are internal angles lying crosswise with parallel lines AD and BC (the bases of the trapezoid are parallel to each other) and a secant line AC, therefore they are equal.
Angles OBC and ODA are equal for the same reason (internal crosswise).

Since all three angles of one triangle are equal to the corresponding angles of another triangle, then these triangles are similar.

What follows from this?

To solve problems in geometry, the similarity of triangles is used as follows. If we know the lengths of two corresponding elements similar triangles, then we find the similarity coefficient (we divide one by the other). From where the lengths of all other elements are related to each other by exactly the same value.

Properties of triangles lying on the lateral side and diagonals of a trapezoid

Consider two triangles lying on the lateral sides of the trapezoid AB and CD. These are triangles AOB and COD. Although the sizes individual parties these triangles may be completely different, but the areas of the triangles formed by the lateral sides and the point of intersection of the diagonals of the trapezoid are equal, that is, the triangles are equal in size.

If we extend the sides of the trapezoid towards the smaller base, then the point of intersection of the sides will be coincide with a straight line that passes through the middle of the bases.

Thus, any trapezoid can be expanded into a triangle. In this case:

• Triangles formed by the bases of a trapezoid with common top at the point of intersection of the extended lateral sides are similar
• The straight line connecting the midpoints of the bases of the trapezoid is, at the same time, the median of the constructed triangle

Properties of a segment connecting the bases of a trapezoid

If you draw a segment whose ends lie on the bases of a trapezoid, which lies at the point of intersection of the diagonals of the trapezoid (KN), then the ratio of its constituent segments from the side of the base to the point of intersection of the diagonals (KO/ON) will be equal to the ratio of the bases of the trapezoid(BC/AD).

KO/ON = BC/AD

This property follows from the similarity of the corresponding triangles (see above).

Properties of a segment parallel to the bases of a trapezoid

If we draw a segment parallel to the bases of the trapezoid and passing through the point of intersection of the trapezoid’s diagonals, then it will have the following properties:

• Specified distance (KM) bisected by the intersection point of the trapezoid's diagonals
• Section length, passing through the point of intersection of the diagonals of the trapezoid and parallel to the bases, is equal KM = 2ab/(a + b)

Formulas for finding the diagonals of a trapezoid

a, b- trapezoid bases

c, d- sides of the trapezoid

d1 d2- diagonals of a trapezoid

α β - angles with a larger base of the trapezoid

Formulas for finding the diagonals of a trapezoid through the bases, sides and angles at the base

The first group of formulas (1-3) reflects one of the main properties of trapezoid diagonals:

1. The sum of the squares of the diagonals of a trapezoid is equal to the sum of the squares of the sides plus twice the product of its bases. This property of trapezoid diagonals can be proven as a separate theorem

2 . This formula obtained by transforming the previous formula. The square of the second diagonal is thrown through the equal sign, after which the square root is extracted from the left and right sides of the expression.

3 . This formula for finding the length of the diagonal of a trapezoid is similar to the previous one, with the difference that another diagonal is left on the left side of the expression

The next group of formulas (4-5) are similar in meaning and express a similar relationship.

The group of formulas (6-7) allows you to find the diagonal of a trapezoid if the larger base of the trapezoid, one side side and the angle at the base are known.

Formulas for finding the diagonals of a trapezoid through height

Task.
The diagonals of the trapezoid ABCD (AD | | BC) intersect at point O. Find the length of the base BC of the trapezoid if the base AD = 24 cm, length AO = 9 cm, length OS = 6 cm.

Solution.
The solution to this problem is ideologically absolutely identical to the previous problems.

Triangles AOD and BOC are similar in three angles - AOD and BOC are vertical, and the remaining angles are pairwise equal, since they are formed by the intersection of one line and two parallel lines.

Since the triangles are similar, all their geometric dimensions are related to each other, just like the geometric dimensions of the segments AO and OC known to us according to the conditions of the problem. That is

AO/OC = AD/BC
9 / 6 = 24 / BC
BC = 24 * 6 / 9 = 16

Answer: 16 cm

Task .
In the trapezoid ABCD it is known that AD=24, BC=8, AC=13, BD=5√17. Find the area of ​​the trapezoid.

Solution .
To find the height of a trapezoid from the vertices of the smaller base B and C, we lower two heights to the larger base. Since the trapezoid is unequal, we denote the length AM = a, length KD = b ( not to be confused with the notation in the formula finding the area of ​​a trapezoid). Since the bases of the trapezoid are parallel, and we dropped two heights perpendicular more reason, then MBCK is a rectangle.

Means
AD = AM+BC+KD
a + 8 + b = 24
a = 16 - b

Triangles DBM and ACK are rectangular, so their right angles are formed by the altitudes of the trapezoid. Let us denote the height of the trapezoid by h. Then, by the Pythagorean theorem

H 2 + (24 - a) 2 = (5√17) 2
And
h 2 + (24 - b) 2 = 13 2

Let's take into account that a = 16 - b, then in the first equation
h 2 + (24 - 16 + b) 2 = 425
h 2 = 425 - (8 + b) 2

Let's substitute the value of the square of the height into the second equation obtained using the Pythagorean Theorem. We get:
425 - (8 + b) 2 + (24 - b) 2 = 169
-(64 + 16b + b) 2 + (24 - b) 2 = -256
-64 - 16b - b 2 + 576 - 48b + b 2 = -256
-64b = -768
b = 12

So KD = 12
Where
h 2 = 425 - (8 + b) 2 = 425 - (8 + 12) 2 = 25
h = 5

Find the area of ​​the trapezoid through its height and half the sum of the bases
, where a b - the base of the trapezoid, h - the height of the trapezoid
S = (24 + 8) * 5 / 2 = 80 cm 2

Answer: the area of ​​the trapezoid is 80 cm2.

In order to feel confident and successfully solve problems in geometry lessons, it is not enough to learn the formulas. They need to be understood first. To be afraid, and even more so to hate formulas, is unproductive. In this article accessible language will be analyzed various ways Finding the area of ​​a trapezoid. To better understand the corresponding rules and theorems, we will pay some attention to its properties. This will help you understand how the rules work and in what cases certain formulas should be applied.

Defining a trapezoid

What kind of figure is this overall? A trapezoid is a polygon with four corners and two parallel sides. The other two sides of the trapezoid can be inclined at different angles. Its parallel sides are called bases, and for non-parallel sides the name “sides” or “hips” is used. Such figures are quite common in everyday life. The contours of the trapezoid can be seen in the silhouettes of clothing, interior items, furniture, dishes and many others. Trapeze happens different types: scalene, equilateral and rectangular. We will examine their types and properties in more detail later in the article.

Properties of a trapezoid

Let us dwell briefly on the properties of this figure. The sum of the angles adjacent to any side is always 180°. It should be noted that all angles of a trapezoid add up to 360°. The trapezoid has the concept of a midline. If you connect the midpoints of the sides with a segment, this will be the middle line. It is designated m. The middle line has important properties: it is always parallel to the bases (we remember that the bases are also parallel to each other) and equal to their half-sum:

This definition must be learned and understood, because it is the key to solving many problems!

With a trapezoid, you can always lower the height to the base. An altitude is a perpendicular, often denoted by the symbol h, that is drawn from any point of one base to another base or its extension. The midline and height will help you find the area of ​​the trapezoid. Similar tasks are the most common in school course geometry and regularly appear among test and examination papers.

The simplest formulas for the area of ​​a trapezoid

Let's look at the two most popular and simple formulas, with the help of which the area of ​​a trapezoid is found. It is enough to multiply the height by half the sum of the bases to easily find what you are looking for:

S = h*(a + b)/2.

In this formula, a, b denote the bases of the trapezoid, h - the height. For ease of perception, in this article, multiplication signs are marked with a symbol (*) in formulas, although in official reference books the multiplication sign is usually omitted.

Let's look at an example.

Given: a trapezoid with two bases equal to 10 and 14 cm, the height is 7 cm. What is the area of ​​the trapezoid?

Let's look at the solution to this problem. Using this formula, you first need to find the half-sum of the bases: (10+14)/2 = 12. So, the half-sum is equal to 12 cm. Now we multiply the half-sum by the height: 12*7 = 84. What we are looking for is found. Answer: The area of ​​the trapezoid is 84 square meters. cm.

Second famous formula states: the area of ​​a trapezoid is equal to the product of the midline and the height of the trapezoid. That is, it actually follows from the previous concept of the middle line: S=m*h.

Using diagonals for calculations

Another way to find the area of ​​a trapezoid is actually not that complicated. It is connected to its diagonals. Using this formula, to find the area, you need to multiply the half-product of its diagonals (d 1 d 2) by the sine of the angle between them:

S = ½ d 1 d 2 sin a.

Let's consider a problem that shows the application of this method. Given: a trapezoid with the length of the diagonals equal to 8 and 13 cm, respectively. The angle a between the diagonals is 30°. Find the area of ​​the trapezoid.

Solution. Using the above formula, it is easy to calculate what is required. As you know, sin 30° is 0.5. Therefore, S = 8*13*0.5=52. Answer: the area is 52 square meters. cm.

Finding the area of ​​an isosceles trapezoid

A trapezoid can be isosceles (isosceles). Its sides are the same and the angles at the bases are equal, which is well illustrated by the figure. Isosceles trapezoid has the same properties as the regular one, plus a number of special ones. A circle can be circumscribed around an isosceles trapezoid, and a circle can be inscribed within it.

What methods are there for calculating the area of ​​such a figure? The method below will require a lot of calculations. To use it, you need to know the values ​​of the sine (sin) and cosine (cos) of the angle at the base of the trapezoid. Their calculations require either Bradis tables or engineering calculator. Here is the formula:

S= c*sin a*(a - c*cos a),

Where With- lateral thigh, a- angle at the lower base.

An equilateral trapezoid has diagonals of equal length. The converse is also true: if a trapezoid has equal diagonals, then it is isosceles. From here following formula, which helps to find the area of ​​a trapezoid - the half product of the square of the diagonals and the sine of the angle between them: S = ½ d 2 sin a.

Finding the area of ​​a rectangular trapezoid

Known special case rectangular trapezoid. This is a trapezoid, in which one side (its thigh) adjoins the bases at a right angle. It has the properties of a regular trapezoid. In addition, she has very interesting feature. The difference in the squares of the diagonals of such a trapezoid is equal to the difference in the squares of its bases. All previously described methods for calculating area are used for it.

We use ingenuity

There is one trick that can help if you forget specific formulas. Let's take a closer look at what a trapezoid is. If we mentally divide it into parts, we will get familiar and understandable geometric shapes: a square or rectangle and a triangle (one or two). If the height and sides of the trapezoid are known, you can use the formulas for the area of ​​a triangle and a rectangle, and then add up all the resulting values.

Let's illustrate this following example. Given a rectangular trapezoid. Angle C = 45°, angles A, D are 90°. The upper base of the trapezoid is 20 cm, the height is 16 cm. You need to calculate the area of ​​the figure.

This figure obviously consists of a rectangle (if two angles are equal to 90°) and a triangle. Since the trapezoid is rectangular, therefore, its height is equal to its side, that is, 16 cm. We have a rectangle with sides of 20 and 16 cm, respectively. Now consider a triangle whose angle is 45°. We know that one side of it is 16 cm. Since this side is also the height of the trapezoid (and we know that the height descends to the base at a right angle), therefore, the second angle of the triangle is 90°. Hence the remaining angle of the triangle is 45°. As a consequence of this we get a rectangular isosceles triangle, whose two sides are the same. This means that the other side of the triangle is equal to the height, that is, 16 cm. All that remains is to calculate the area of ​​the triangle and the rectangle and add the resulting values.

The area of ​​a right triangle is equal to half the product of its legs: S = (16*16)/2 = 128. The area of ​​a rectangle is equal to the product of its width and length: S = 20*16 = 320. We found the required: area of ​​the trapezoid S = 128 + 320 = 448 sq. see. You can easily double-check yourself using the above formulas, the answer will be identical.

We use the Pick formula

S = M/2 + N - 1,

in this formula M is the number of nodes, i.e. intersections of the lines of the figure with the lines of the cell at the boundaries of the trapezoid (orange dots in the figure), N is the number of nodes inside the figure (blue dots). It is most convenient to use it when finding the area irregular polygon. However, the larger the arsenal of techniques used, the fewer errors and better the results.

Of course, the information provided does not exhaust the types and properties of a trapezoid, as well as methods for finding its area. This article provides an overview of its most important characteristics. When solving geometric problems, it is important to act gradually, start with easy formulas and problems, consistently consolidate your understanding, and move to another level of complexity.

Collected together the most common formulas will help students navigate the various ways to calculate the area of ​​a trapezoid and better prepare for tests and tests on this topic.