If the diagonals in an isosceles trapezoid are perpendicular, the following theoretical material will be useful in solving the problem.

1. If the diagonals in an isosceles trapezoid are perpendicular, the height of the trapezoid is equal to half the sum of the bases.

Let us draw a line CF parallel to BD through point C and extend the line AD until it intersects with CF.

Quadrilateral BCFD is a parallelogram (BC∥ DF as the base of a trapezoid, BD∥ CF by construction). So CF=BD, DF=BC and AF=AD+BC.

Triangle ACF is right-angled (if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other line). Since in an isosceles trapezoid the diagonals are equal, and CF = BD, then CF = AC, that is, triangle ACF is isosceles with base AF. This means that its height CN is also the median. And since the median of a right triangle drawn to the hypotenuse is equal to half of it, then

what in general view can be written as

where h is the height of the trapezoid, a and b are its bases.

2. If the diagonals in an isosceles trapezoid are perpendicular, then its height is equal to the midline.

Since the midline of the trapezoid m is equal to half the sum of the bases, then

3. If the diagonals in an isosceles trapezoid are perpendicular, then the area of ​​the trapezoid is equal to the square of the height of the trapezoid (or the square of the half-sum of the bases, or the square of the midline).

Since the area of ​​a trapezoid is found by the formula

and the height, half the sum of the bases and the middle line of an isosceles trapezoid with perpendicular diagonals are equal to each other:

Trapezoid is special case a quadrilateral in which one pair of sides is parallel. The term "trapezoid" comes from Greek wordτράπεζα, meaning “table”, “table”. In this article we will look at the types of trapezoid and its properties. In addition, we will figure out how to calculate individual elements of this For example, the diagonal of an isosceles trapezoid, the center line, area, etc. The material is presented in the style of elementary popular geometry, i.e. in an easily accessible form.

General information

First, let's figure out what a quadrilateral is. This figure is a special case of a polygon containing four sides and four vertices. Two vertices of a quadrilateral that are not adjacent are called opposite. The same can be said about two non-adjacent sides. The main types of quadrilaterals are parallelogram, rectangle, rhombus, square, trapezoid and deltoid.

So let's get back to trapezoids. As we have already said, this figure has two parallel sides. They are called bases. The other two (non-parallel) are the lateral sides. In exam materials and various tests very often you can find problems related to trapezoids, the solution of which often requires the student to have knowledge not provided for in the program. The school geometry course introduces students to the properties of angles and diagonals, as well as the midline of an isosceles trapezoid. But, in addition to this, the mentioned geometric figure has other features. But more about them a little later...

Types of trapezoid

There are many types of this figure. However, most often it is customary to consider two of them - isosceles and rectangular.

1. A rectangular trapezoid is a figure in which one of the sides is perpendicular to the bases. Her two angles are always equal to ninety degrees.

2. An isosceles trapezoid is a geometric figure whose sides are equal to each other. This means that the angles at the bases are also equal in pairs.

The main principles of the methodology for studying the properties of a trapezoid

The main principle includes the use of the so-called task approach. In fact, there is no need to introduce new properties of this figure into the theoretical course of geometry. They can be discovered and formulated in the process of solving various problems (preferably system ones). At the same time, it is very important that the teacher knows what tasks need to be assigned to students at one time or another during the educational process. Moreover, each property of a trapezoid can be represented as a key task in the task system.

The second principle is the so-called spiral organization of the study of the “remarkable” properties of the trapezoid. This implies a return in the learning process to individual features of a given geometric figure. This makes it easier for students to remember them. For example, the property of four points. It can be proven both when studying similarity and subsequently using vectors. And the equivalence of triangles adjacent to the lateral sides of a figure can be proven by applying not only the properties of triangles with equal heights drawn to the sides that lie on the same straight line, but also using the formula S = 1/2(ab*sinα). In addition, you can work on an inscribed trapezoid or a right triangle on an inscribed trapezoid, etc.

The use of “extracurricular” features of a geometric figure in the content of a school course is a task-based technology for teaching them. Constantly referring to the properties being studied while going through other topics allows students to gain a deeper knowledge of the trapezoid and ensures the success of solving assigned problems. So, let's start studying this wonderful figure.

Elements and properties of an isosceles trapezoid

As we have already noted, this geometric figure has equal sides. It is also known as the correct trapezoid. Why is it so remarkable and why did it get such a name? The peculiarity of this figure is that not only the sides and angles at the bases are equal, but also the diagonals. In addition, the sum of the angles of an isosceles trapezoid is 360 degrees. But that's not all! Of all the known trapezoids, only an isosceles one can be described as a circle. This is due to the fact that the sum of the opposite angles of this figure is equal to 180 degrees, and only under this condition can one describe a circle around the quadrilateral. The next property of the geometric figure under consideration is that the distance from the vertex of the base to the projection of the opposite vertex onto the straight line that contains this base will be equal to the midline.

Now let's figure out how to find the angles of an isosceles trapezoid. Let us consider a solution to this problem, provided that the dimensions of the sides of the figure are known.

Solution

Typically, a quadrilateral is usually denoted by the letters A, B, C, D, where BS and AD are the bases. In an isosceles trapezoid, the sides are equal. We will assume that their size is equal to X, and the sizes of the bases are equal to Y and Z (smaller and larger, respectively). To carry out the calculation, it is necessary to draw the height H from angle B. The result is a right triangle ABN, where AB is the hypotenuse, and BN and AN are the legs. We calculate the size of the leg AN: we subtract the smaller one from the larger base, and divide the result by 2. We write it in the form of a formula: (Z-Y)/2 = F. Now, to calculate the acute angle of the triangle, we use the cos function. We get the following entry: cos(β) = X/F. Now we calculate the angle: β=arcos (X/F). Further, knowing one angle, we can determine the second, for this we perform an elementary arithmetic operation: 180 - β. All angles are defined.

There is a second solution to this problem. First, we lower it from the corner to height H. We calculate the value of the leg BN. We know that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. We get: BN = √(X2-F2). Next we use trigonometric function tg. As a result, we have: β = arctan (BN/F). An acute angle has been found. Next, we determine the obtuse angle similarly to the first method.

Property of diagonals of an isosceles trapezoid

First, let's write down four rules. If the diagonals in an isosceles trapezoid are perpendicular, then:

The height of the figure will be equal to the sum of the bases divided by two;

Its height and midline are equal;

The center of the circle is the point at which ;

If the lateral side is divided by the point of tangency into segments H and M, then it is equal to square root products of these segments;

The quadrilateral that is formed by the points of tangency, the vertex of the trapezoid and the center of the inscribed circle is a square whose side is equal to the radius;

The area of ​​a figure is equal to the product of the bases and the product of half the sum of the bases and its height.

Similar trapezoids

This topic is very convenient for studying the properties of this For example, the diagonals divide a trapezoid into four triangles, and those adjacent to the bases are similar, and those adjacent to the sides are equal in size. This statement can be called a property of the triangles into which the trapezoid is divided by its diagonals. The first part of this statement is proven through the sign of similarity at two angles. To prove the second part, it is better to use the method given below.

Proof of the theorem

We accept that the figure ABSD (AD and BS are the bases of the trapezoid) is divided by diagonals VD and AC. The point of their intersection is O. We get four triangles: AOS - at the lower base, BOS - at the upper base, ABO and SOD at the sides. Triangles SOD and BOS have a common height if the segments BO and OD are their bases. We find that the difference between their areas (P) is equal to the difference between these segments: PBOS/PSOD = BO/OD = K. Therefore, PSOD = PBOS/K. Similarly, triangles BOS and AOB have a common height. We take the segments CO and OA as their bases. We get PBOS/PAOB = CO/OA = K and PAOB = PBOS/K. It follows from this that PSOD = PAOB.

To consolidate the material, students are recommended to find the connection between the areas of the resulting triangles into which the trapezoid is divided by its diagonals by solving the following problem. It is known that triangles BOS and AOD have equal areas; it is necessary to find the area of ​​the trapezoid. Since PSOD = PAOB, it means PABSD = PBOS+PAOD+2*PSOD. From the similarity of triangles BOS and AOD it follows that BO/OD = √(PBOS/PAOD). Therefore, PBOS/PSOD = BO/OD = √(PBOS/PAOD). We get PSOD = √(PBOS*PAOD). Then PABSD = PBOS+PAOD+2*√(PBOS*PAOD) = (√PBOS+√PAOD)2.

Properties of similarity

Continuing to develop this topic, one can prove other interesting features trapezoid. Thus, using similarity, one can prove the property of a segment that passes through the point formed by the intersection of the diagonals of this geometric figure, parallel to the bases. To do this, let's solve the following problem: we need to find the length of the segment RK that passes through point O. From the similarity of triangles AOD and BOS it follows that AO/OS = AD/BS. From the similarity of triangles AOP and ASB it follows that AO/AC=RO/BS=AD/(BS+AD). From here we get that RO=BS*BP/(BS+BP). Similarly, from the similarity of triangles DOC and DBS, it follows that OK = BS*AD/(BS+AD). From here we get that RO=OK and RK=2*BS*AD/(BS+AD). A segment passing through the point of intersection of the diagonals, parallel to the bases and connecting two lateral sides, is divided in half by the point of intersection. Its length is the harmonic mean of the figure's bases.

Consider the following property of a trapezoid, which is called the property of four points. The intersection points of the diagonals (O), the intersection of the continuation of the sides (E), as well as the midpoints of the bases (T and F) always lie on the same line. This can be easily proven by the similarity method. The resulting triangles BES and AED are similar, and in each of them the medians ET and EJ divide the vertex angle E into equal parts. Therefore, points E, T and F lie on the same straight line. In the same way, points T, O, and Zh are located on the same straight line. All this follows from the similarity of triangles BOS and AOD. From here we conclude that all four points - E, T, O and F - will lie on the same straight line.

Using similar trapezoids, you can ask students to find the length of the segment (LS) that divides the figure into two similar ones. This segment must be parallel to the bases. Since the resulting trapezoids ALFD and LBSF are similar, then BS/LF = LF/AD. It follows that LF=√(BS*AD). We find that the segment dividing the trapezoid into two similar ones has a length equal to the geometric mean of the lengths of the bases of the figure.

Let's consider next property similarities It is based on a segment that divides the trapezoid into two equal figures. We assume that the trapezoid ABSD is divided by the segment EH into two similar ones. From vertex B a height is omitted, which is divided by segment EN into two parts - B1 and B2. We get: PABSD/2 = (BS+EN)*B1/2 = (AD+EN)*B2/2 and PABSD = (BS+AD)*(B1+B2)/2. Next, we compose a system whose first equation is (BS+EN)*B1 = (AD+EN)*B2 and the second (BS+EN)*B1 = (BS+AD)*(B1+B2)/2. It follows that B2/B1 = (BS+EN)/(AD+EN) and BS+EN = ((BS+AD)/2)*(1+B2/B1). We find that the length of the segment dividing the trapezoid into two equal ones is equal to the root mean square of the lengths of the bases: √((BS2+AD2)/2).

Similarity findings

Thus, we have proven that:

1. The segment connecting the midpoints of the lateral sides of a trapezoid is parallel to AD and BS and is equal to the arithmetic mean of BS and AD (the length of the base of the trapezoid).

2. The line passing through the point O of the intersection of the diagonals parallel to AD and BS will be equal to the harmonic mean of the numbers AD and BS (2*BS*AD/(BS+AD)).

3. The segment dividing the trapezoid into similar ones has the length of the geometric mean of the bases BS and AD.

4. An element dividing a figure into two equal ones has the length of the root mean square of the numbers AD and BS.

To consolidate the material and understand the connection between the considered segments, the student needs to construct them for a specific trapezoid. He can easily display the middle line and the segment that passes through point O - the intersection of the diagonals of the figure - parallel to the bases. But where will the third and fourth be located? This answer will lead the student to the discovery of the desired relationship between average values.

A segment connecting the midpoints of the diagonals of a trapezoid

Consider the following property of this figure. We assume that the segment MH is parallel to the bases and bisects the diagonals. Let's call the intersection points Ш and Ш. This segment will be equal to half the difference of the bases. Let's look at this in more detail. MS is the middle line of the ABS triangle, it is equal to BS/2. MSH is the middle line of the triangle ABD, it is equal to AD/2. Then we get that ShShch = MSh-MSh, therefore, ShShch = AD/2-BS/2 = (AD+VS)/2.

Center of gravity

Let's look at how this element is determined for a given geometric figure. To do this, it is necessary to extend the grounds in opposite sides. What does it mean? You need to add the lower base to the upper base - in any direction, for example, to the right. And we extend the lower one by the length of the upper one to the left. Next, we connect them diagonally. The point of intersection of this segment with the midline of the figure is the center of gravity of the trapezoid.

Inscribed and circumscribed trapezoids

Let's list the features of such figures:

1. A trapezoid can be inscribed in a circle only if it is isosceles.

2. A trapezoid can be described around a circle, provided that the sum of the lengths of their bases is equal to the sum of the lengths of the sides.

Corollaries of the incircle:

1. The height of the described trapezoid is always equal to two radii.

2. The side of the described trapezoid is observed from the center of the circle at a right angle.

The first corollary is obvious, but to prove the second it is necessary to establish that the angle SOD is right, which, in fact, is also not difficult. But knowledge of this property will allow you to use a right triangle when solving problems.

Now let us specify these consequences for an isosceles trapezoid inscribed in a circle. We find that the height is the geometric mean of the bases of the figure: H=2R=√(BS*AD). While practicing the basic technique for solving problems for trapezoids (the principle of drawing two heights), the student must solve the following task. We assume that BT is the height of the isosceles figure ABSD. It is necessary to find the segments AT and TD. Using the formula described above, this will not be difficult to do.

Now let's figure out how to determine the radius of a circle using the area of ​​the circumscribed trapezoid. We lower the height from vertex B to the base AD. Since the circle is inscribed in a trapezoid, then BS+AD = 2AB or AB = (BS+AD)/2. From triangle ABN we find sinα = BN/AB = 2*BN/(BS+AD). PABSD = (BS+BP)*BN/2, BN=2R. We get PABSD = (BS+BP)*R, it follows that R = PABSD/(BS+BP).

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All formulas for the midline of a trapezoid

Now it's time to move on to the last element of this geometric figure. Let's figure out what the middle line of the trapezoid (M) is equal to:

1. Through the bases: M = (A+B)/2.

2. Through height, base and corners:

M = A-H*(ctgα+ctgβ)/2;

M = B+N*(ctgα+ctgβ)/2.

3. Through height, diagonals and the angle between them. For example, D1 and D2 are the diagonals of a trapezoid; α, β - angles between them:

M = D1*D2*sinα/2Н = D1*D2*sinβ/2Н.

4. Through area and height: M = P/N.

Before we figure out how to find the diagonal of a trapezoid, let's remember what a trapezoid is. In planimetry, a trapezoid is a quadrilateral whose two opposite sides are parallel to each other. These parallel sides are called the bases of the trapezoid, and the rest are called the lateral sides. The sides can be the same, then we are dealing with an isosceles trapezoid.

Next, we will analyze in detail the procedure for finding the length of the diagonals for the general case - an isosceles trapezoid. In this case, we will proceed from the fact that the initial data are the lengths of all four sides of the trapezoid, the angles at the base are unknown.

Calculation of the diagonal of a trapezoid

The trapezoid ABCD shown in the figure has two diagonals AC and BD. The order of finding their lengths is the same, so let’s consider everything using the example of finding the diagonal BD opposite ˂BAD.

Diagonal BD is also a side of triangle ABD and can be calculated using the law of cosines using the formula:

BD = √(AB 2 +AD 2 -2AB . AD . cos ˂BAD)

In this formula we know all the quantities except the cosine ˂BAD. To calculate it, we will need to perform a small transformation on the drawing. Let’s “cut” a BNMC rectangle from the original trapezoid. As a result, we obtain a triangle ABD", in which the side BD" will be equal to the side of the trapezoid CD.

˂BAD" in a triangle is equal to ˂BAD in a trapezoid, since we did not perform any transformations with triangle ABN. So, in this triangle ABD" side AB is known to us, side BD" = CD, and side AD" = AD - NM = AD – B.C.

It turns out that by the cosine theorem cos ˂BAD = cos ˂BAD" = (AB 2 + AD" 2 – BD" 2)/2AB. AD" = (AB 2 + (AD – BC) 2 – CD 2)/2AB. (AD–BC)

Now substituting the resulting expression into the formula found earlier, we get:

BD = √(AB 2 +AD 2 -2AB . AD . cos ˂BAD) = √(AB 2 +AD 2 -2AB . AD . (AB 2 +(AD – BC) 2 – CD 2)/2AB . (AD – BC)) = √(AB 2 + AD 2 – AD. (AB 2 +(AD – BC) 2 – CD 2)/(AD – BC)) = √(AB 2 + AD 2 – AD. (AD – BC) 2 /(AD – BC) – AD . (AB 2 – CD 2)/(AD – BC)) = √(AB 2 + AD 2 – AD 2 + AD . BC – AD . (AB 2 – CD 2 )/(AD – BC)) = √(AB 2 + AD . BC – AD . (AB 2 – CD 2)/(AD – BC))

BD = √(AB 2 + AD . BC – AD . (AB 2 – CD 2)/(AD – BC))

The resulting formula for the diagonal of a trapezoid is valid for any values ​​of the lengths of the sides of the original quadrilateral.

For the second diagonal, the formula will accordingly take the form:

AC = √(CD 2 + AD . BC – AD . (CD 2 – AB 2)/(AD – BC))

Diagonal of an isosceles trapezoid

If you are interested in how to find the diagonal of an isosceles trapezoid, the resulting formula can be greatly simplified. Indeed, in an isosceles trapezoid AB = CD, therefore AB 2 – CD 2 = 0 and the formula for the length of the diagonal is reduced to the form:

BD = √(AB 2 + AD . BC)

The diagonals of an isosceles trapezoid are equal to each other, so the second diagonal is found using the same formula.

If the initial data are the lengths of the bases of the trapezoid, one of the lateral sides and the angles at the base, then the task of finding the diagonal of the trapezoid is reduced to calculating the side of the triangle using the cosine theorem.

(from ancient Greek τραπέζιον - “table”; τράπεζα - “table, food”) - a quadrilateral with only one pair of opposite sides parallel.

Sometimes a trapezoid is defined as a quadrilateral in which a pair of opposite sides are parallel (the other is not specified), in this case it is a special case of a trapezoid. In particular, there is the concept of a curvilinear trapezoid.

Trapezoid elements

• Parallel sides are called reasons trapezoids.
• The other two sides are called sides.
• connecting the midpoints of the lateral sides is called midline trapezoids.
• The distance between the bases is called height trapezoids.

Types of trapezoids

• A trapezoid whose sides are equal is called isosceles.
• A trapezoid in which one of the angles is “straight” is called rectangular.

General properties

• The midline of the trapezoid is parallel to the bases and equal to their half-sum.
• The segment connecting the midpoints of the diagonals is equal to half the difference of the bases.
• (Generalized Thales). Parallel lines intersecting the sides of an angle cut off proportional segments from the sides of the angle.

Properties of an isosceles trapezoid

• , passing through the midpoints of the bases, is perpendicular to the bases and is the axis of symmetry of the trapezoid.
• The height lowered from the top to the larger base divides it into two segments, one of which is equal to half the sum of the bases, the other - half the difference of the bases.
• In an isosceles trapezoid, the angles at any base are equal.
• In an isosceles trapezoid, the lengths of the diagonals are equal.
• About an isosceles trapezoid can be described.
• If the diagonals in an isosceles trapezoid are perpendicular, then the height is equal to half the sum of the bases,

A quadrilateral in which only two sides are parallel is called trapezoid.

The parallel sides of a trapezoid are called its reasons, and those sides that are not parallel are called sides. If the sides are equal, then such a trapezoid is isosceles. The distance between the bases is called the height of the trapezoid.

Middle Line Trapezoid

The midline is a segment connecting the midpoints of the lateral sides of the trapezoid. The midline of the trapezoid is parallel to its bases.

Theorem:

If a straight line intersecting the middle of one side is parallel to the bases of the trapezoid, then it bisects the second side trapezoids.

Theorem:

The length of the middle line is equal to the arithmetic mean of the lengths of its bases

MN midline, AB and CD - bases, AD and BC - lateral sides

MN = (AB + DC)/2

Theorem:

The length of the midline of a trapezoid is equal to the arithmetic mean of the lengths of its bases.

The main task: Prove that the midline of a trapezoid bisects a segment whose ends lie in the middle of the bases of the trapezoid.

Middle Line of the Triangle

The segment connecting the midpoints of two sides of a triangle is called the midline of the triangle. It is parallel to the third side and its length is equal to half the length of the third side.
Theorem: If a line intersecting the midpoint of one side of a triangle is parallel to the other side of the triangle, then it bisects the third side.

AM = MC and BN = NC =>

Applying the midline properties of a triangle and trapezoid

Dividing a segment into a certain number of equal parts.
Task: Divide segment AB into 5 equal parts.
Solution:
Let p be a random ray whose origin is point A and which does not lie on line AB. We sequentially set aside 5 equal segments on p AA 1 = A 1 A 2 = A 2 A 3 = A 3 A 4 = A 4 ​​A 5
We connect A 5 to B and draw such lines through A 4, A 3, A 2 and A 1 that are parallel to A 5 B. They intersect AB respectively at points B 4, B 3, B 2 and B 1. These points divide segment AB into 5 equal parts. Indeed, from the trapezoid BB 3 A 3 A 5 we see that BB 4 = B 4 B 3. In the same way, from the trapezoid B 4 B 2 A 2 A 4 we obtain B 4 B 3 = B 3 B 2

While from the trapezoid B 3 B 1 A 1 A 3, B 3 B 2 = B 2 B 1.
Then from B 2 AA 2 it follows that B 2 B 1 = B 1 A. In conclusion we get:
AB 1 = B 1 B 2 = B 2 B 3 = B 3 B 4 = B 4 B
It is clear that to divide the segment AB into another number of equal parts, we need to project the same number of equal segments onto the ray p. And then continue in the manner described above.