Theorem on the property of the midline of a trapezoid. Middle line

Trapezoid is a quadrilateral that has two parallel sides, which are the bases, and two non-parallel sides, which are the sides.

There are also names such as isosceles or equilateral.

is a trapezoid whose side angles are right.

Trapezoid elements

a, b - trapezoid bases(a parallel to b),

m, n - sides trapezoids,

d 1 , d 2 — diagonals trapezoids,

h - height trapezoid (a segment connecting the bases and at the same time perpendicular to them),

MN - midline(segment connecting the midpoints of the sides).

Area of trapezoid

Through the half-sum of bases a, b and height h: S = \frac(a + b)(2)\cdot h
Through the center line MN and height h: S = MN\cdot h
Through diagonals d 1, d 2 and the angle (\sin \varphi) between them: S = \frac(d_(1) d_(2) \sin \varphi)(2)

Properties of a trapezoid

Midline of trapezoid

Middle line parallel to the bases, equal to their half-sum and divides each segment with ends located on straight lines that contain the bases (for example, the height of the figure) in half:

MN || a, MN || b, MN = \frac(a + b)(2)

Sum of trapezoid angles

Sum of trapezoid angles, adjacent to each side, is equal to 180^(\circ) :

\alpha + \beta = 180^(\circ)

\gamma + \delta =180^(\circ)

Equal-area trapezoid triangles

Equal in size, that is, having equal areas, are the diagonal segments and triangles AOB and DOC formed by the lateral sides.

The similarity of the formed trapezoid triangles

Similar triangles are AOD and COB, which are formed by their bases and diagonal segments.

\triangle AOD \sim \triangle COB

Similarity coefficient k is found by the formula:

k = \frac(AD)(BC)

Moreover, the ratio of the areas of these triangles is equal to k^(2) .

Ratio of lengths of segments and bases

Each segment connecting the bases and passing through the point of intersection of the diagonals of the trapezoid is divided by this point in the ratio:

\frac(OX)(OY) = \frac(BC)(AD)

This will also be true for the height with the diagonals themselves.

In this article we will try to reflect the properties of a trapezoid as fully as possible. In particular, we will talk about general signs and properties of a trapezoid, as well as about the properties of an inscribed trapezoid and about a circle inscribed in a trapezoid. We will also touch on the properties of an isosceles and rectangular trapezoid.

An example of solving a problem using the properties discussed will help you sort it into places in your head and better remember the material.

Trapeze and all-all-all

To begin with, let us briefly recall what a trapezoid is and what other concepts are associated with it.

So, a trapezoid is a quadrilateral figure, two of whose sides are parallel to each other (these are the bases). And the two are not parallel - these are the sides.

In a trapezoid, the height can be lowered - perpendicular to the bases. The center line and diagonals are drawn. It is also possible to draw a bisector from any angle of the trapezoid.

We will now talk about the various properties associated with all these elements and their combinations.

Properties of trapezoid diagonals

To make it clearer, while you are reading, sketch out the trapezoid ACME on a piece of paper and draw diagonals in it.

If you find the midpoints of each of the diagonals (let's call these points X and T) and connect them, you get a segment. One of the properties of the diagonals of a trapezoid is that the segment HT lies on the midline. And its length can be obtained by dividing the difference of the bases by two: ХТ = (a – b)/2.
Before us is the same trapezoid ACME. The diagonals intersect at point O. Let's look at the triangles AOE and MOK, formed by segments of the diagonals together with the bases of the trapezoid. These triangles are similar. The similarity coefficient k of triangles is expressed through the ratio of the bases of the trapezoid: k = AE/KM.
The ratio of the areas of triangles AOE and MOK is described by the coefficient k 2 .
The same trapezoid, the same diagonals intersecting at point O. Only this time we will consider the triangles that the segments of the diagonals formed together with the sides of the trapezoid. The areas of triangles AKO and EMO are equal in size - their areas are the same.
Another property of a trapezoid involves the construction of diagonals. So, if you continue the sides of AK and ME in the direction of the smaller base, then sooner or later they will intersect at a certain point. Next, draw a straight line through the middle of the bases of the trapezoid. It intersects the bases at points X and T.
If we now extend the line XT, then it will connect together the point of intersection of the diagonals of the trapezoid O, the point at which the extensions of the sides and the middle of the bases X and T intersect.
Through the point of intersection of the diagonals we will draw a segment that will connect the bases of the trapezoid (T lies on the smaller base KM, X on the larger AE). The intersection point of the diagonals divides this segment in the following ratio: TO/OX = KM/AE.
Now, through the point of intersection of the diagonals, we will draw a segment parallel to the bases of the trapezoid (a and b). The intersection point will divide it into two equal parts. You can find the length of the segment using the formula 2ab/(a + b).

Properties of the midline of a trapezoid

Draw the middle line in the trapezoid parallel to its bases.

The length of the midline of a trapezoid can be calculated by adding the lengths of the bases and dividing them in half: m = (a + b)/2.
If you draw any segment (height, for example) through both bases of the trapezoid, the middle line will divide it into two equal parts.

Trapezoid Bisector Property

Select any angle of the trapezoid and draw a bisector. Let's take, for example, the angle KAE of our trapezoid ACME. Having completed the construction yourself, you can easily verify that the bisector cuts off from the base (or its continuation on a straight line outside the figure itself) a segment of the same length as the side.

Properties of trapezoid angles

Whichever of the two pairs of angles adjacent to the side you choose, the sum of the angles in the pair is always 180 0: α + β = 180 0 and γ + δ = 180 0.
Let's connect the midpoints of the bases of the trapezoid with a segment TX. Now let's look at the angles at the bases of the trapezoid. If the sum of the angles for any of them is 90 0, the length of the segment TX can be easily calculated based on the difference in the lengths of the bases, divided in half: TX = (AE – KM)/2.
If parallel lines are drawn through the sides of a trapezoid angle, they will divide the sides of the angle into proportional segments.

Properties of an isosceles (equilateral) trapezoid

In an isosceles trapezoid, the angles at any base are equal.
Now build a trapezoid again to make it easier to imagine what we're talking about. Look carefully at the base AE - the vertex of the opposite base M is projected to a certain point on the line that contains AE. The distance from vertex A to the projection point of vertex M and the middle line of the isosceles trapezoid are equal.
A few words about the property of the diagonals of an isosceles trapezoid - their lengths are equal. And also the angles of inclination of these diagonals to the base of the trapezoid are the same.
Only around an isosceles trapezoid can a circle be described, since the sum of the opposite angles of a quadrilateral is 180 0 - a prerequisite for this.
The property of an isosceles trapezoid follows from the previous paragraph - if a circle can be described near the trapezoid, it is isosceles.
From the features of an isosceles trapezoid follows the property of the height of a trapezoid: if its diagonals intersect at right angles, then the length of the height is equal to half the sum of the bases: h = (a + b)/2.
Again, draw the segment TX through the midpoints of the bases of the trapezoid - in an isosceles trapezoid it is perpendicular to the bases. And at the same time TX is the axis of symmetry of an isosceles trapezoid.
This time, lower the height from the opposite vertex of the trapezoid onto the larger base (let's call it a). You will get two segments. The length of one can be found if the lengths of the bases are added and divided in half: (a + b)/2. We get the second one when we subtract the smaller one from the larger base and divide the resulting difference by two: (a – b)/2.

Properties of a trapezoid inscribed in a circle

Since we are already talking about a trapezoid inscribed in a circle, let us dwell on this issue in more detail. In particular, on where the center of the circle is in relation to the trapezoid. Here, too, it is recommended that you take the time to pick up a pencil and draw what will be discussed below. This way you will understand faster and remember better.

The location of the center of the circle is determined by the angle of inclination of the diagonal of the trapezoid to its side. For example, the diagonal may extend from the top of the trapezoid at right angles to the side. In this case, the larger base intersects the center of the circumscribed circle exactly in the middle (R = ½AE).
The diagonal and the side can also meet at an acute angle - then the center of the circle is inside the trapezoid.
The center of the circumscribed circle may be outside the trapezoid, beyond its larger base, if there is an obtuse angle between the diagonal of the trapezoid and the side.
The angle formed by the diagonal and the large base of the trapezoid ACME (inscribed angle) is half the central angle that corresponds to it: MAE = ½MOE.
Briefly about two ways to find the radius of a circumscribed circle. Method one: look carefully at your drawing - what do you see? You can easily notice that the diagonal splits the trapezoid into two triangles. The radius can be found by the ratio of the side of the triangle to the sine of the opposite angle, multiplied by two. For example, R = AE/2*sinAME. The formula can be written in a similar way for any of the sides of both triangles.
Method two: find the radius of the circumscribed circle through the area of the triangle formed by the diagonal, side and base of the trapezoid: R = AM*ME*AE/4*S AME.

Properties of a trapezoid circumscribed about a circle

You can fit a circle into a trapezoid if one condition is met. Read more about it below. And together this combination of figures has a number of interesting properties.

If a circle is inscribed in a trapezoid, the length of its midline can be easily found by adding the lengths of the sides and dividing the resulting sum in half: m = (c + d)/2.
For the trapezoid ACME, described about a circle, the sum of the lengths of the bases is equal to the sum of the lengths of the sides: AK + ME = KM + AE.
From this property of the bases of a trapezoid, the converse statement follows: a circle can be inscribed in a trapezoid whose sum of bases is equal to the sum of its sides.
The tangent point of a circle with radius r inscribed in a trapezoid divides the side into two segments, let's call them a and b. The radius of a circle can be calculated using the formula: r = √ab.
And one more property. To avoid confusion, draw this example yourself too. We have the good old trapezoid ACME, described around a circle. It contains diagonals that intersect at point O. The triangles AOK and EOM formed by the segments of the diagonals and the lateral sides are rectangular.
The heights of these triangles, lowered to the hypotenuses (i.e., the lateral sides of the trapezoid), coincide with the radii of the inscribed circle. And the height of the trapezoid coincides with the diameter of the inscribed circle.

Properties of a rectangular trapezoid

A trapezoid is called rectangular if one of its angles is right. And its properties stem from this circumstance.

A rectangular trapezoid has one of its sides perpendicular to its base.
Height and lateral side of the trapezoid adjacent to right angle, are equal. This allows you to calculate the area of a rectangular trapezoid ( general formula S = (a + b) * h/2) not only through the height, but also through the side adjacent to the right angle.
For a rectangular trapezoid, the general properties of the diagonals of a trapezoid already described above are relevant.

Evidence of some properties of the trapezoid

Equality of angles at the base of an isosceles trapezoid:

You probably already guessed that here we will need the AKME trapezoid again - draw an isosceles trapezoid. Draw a straight line MT from vertex M, parallel to the side of AK (MT || AK).

The resulting quadrilateral AKMT is a parallelogram (AK || MT, KM || AT). Since ME = KA = MT, ∆ MTE is isosceles and MET = MTE.

AK || MT, therefore MTE = KAE, MET = MTE = KAE.

Where does AKM = 180 0 - MET = 180 0 - KAE = KME.

Q.E.D.

Now, based on the property of an isosceles trapezoid (equality of diagonals), we prove that trapezoid ACME is isosceles:

First, let’s draw a straight line MX – MX || KE. We obtain a parallelogram KMHE (base – MX || KE and KM || EX).

∆AMX is isosceles, since AM = KE = MX, and MAX = MEA.

MH || KE, KEA = MXE, therefore MAE = MXE.

It turned out that the triangles AKE and EMA are equal to each other, since AM = KE and AE are the common side of the two triangles. And also MAE = MXE. We can conclude that AK = ME, and from this it follows that the trapezoid AKME is isosceles.

Review task

The bases of the trapezoid ACME are 9 cm and 21 cm, the side side KA, equal to 8 cm, forms an angle of 150 0 with the smaller base. You need to find the area of the trapezoid.

Solution: From vertex K we lower the height to the larger base of the trapezoid. And let's start looking at the angles of the trapezoid.

Angles AEM and KAN are one-sided. This means that in total they give 180 0. Therefore, KAN = 30 0 (based on the property of trapezoidal angles).

Let us now consider the rectangular ∆ANC (I believe this point is obvious to readers without additional evidence). From it we will find the height of the trapezoid KH - in a triangle it is the leg that lies opposite the angle of 30 0. Therefore, KH = ½AB = 4 cm.

We find the area of the trapezoid using the formula: S ACME = (KM + AE) * KN/2 = (9 + 21) * 4/2 = 60 cm 2.

Afterword

If you carefully and thoughtfully studied this article, were not too lazy to draw trapezoids for all the given properties with a pencil in your hands and analyze them in practice, you should have mastered the material well.

Of course, there is a lot of information here, varied and sometimes even confusing: it is not so difficult to confuse the properties of the described trapezoid with the properties of the inscribed one. But you yourself have seen that the difference is huge.

Now you have a detailed summary of all general properties trapezoids. As well as specific properties and characteristics of isosceles and rectangular trapezoids. It is very convenient to use to prepare for tests and exams. Try it yourself and share the link with your friends!

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