Signs of the midline of the trapezium. Median line of the trapezoid

The concept of the midline of the trapezoid

First, let's remember what figure is called a trapezoid.

Definition 1

A trapezoid is a quadrilateral in which two sides are parallel and the other two are not parallel.

In this case, parallel sides are called the bases of the trapezoid, and not parallel - the sides of the trapezoid.

Definition 2

The midline of a trapezoid is a line segment that connects the midpoints of the sides of the trapezoid.

Trapezium midline theorem

We now introduce the theorem on the midline of a trapezoid and prove it by the vector method.

Theorem 1

The median line of the trapezoid is parallel to the bases and equal to half their sum.

Proof.

Let us be given a trapezoid $ABCD$ with bases $AD\ and\ BC$. And let $MN$ be the midline of this trapezoid (Fig. 1).

Figure 1. The middle line of the trapezoid

Let us prove that $MN||AD\ and\ MN=\frac(AD+BC)(2)$.

Consider the vector $\overrightarrow(MN)$. Next, we use the polygon rule for vector addition. On the one hand, we get that

On the other side

Adding the last two equalities, we get

Since $M$ and $N$ are the midpoints of the sides of the trapezoid, we have

We get:

Hence

From the same equality (since $\overrightarrow(BC)$ and $\overrightarrow(AD)$ are codirectional and, therefore, collinear), we get that $MN||AD$.

The theorem has been proven.

Examples of tasks on the concept of the midline of a trapezoid

Example 1

The sides of the trapezoid are $15\cm$ and $17\cm$ respectively. The perimeter of the trapezoid is $52\cm$. Find the length of the midline of the trapezoid.

Decision.

Denote the midline of the trapezoid by $n$.

The sum of the sides is

Therefore, since the perimeter is $52\ cm$, the sum of the bases is

Hence, by Theorem 1, we obtain

Answer:$10\cm$.

Example 2

The ends of the circle's diameter are $9$ cm and $5$ cm respectively from its tangent. Find the diameter of this circle.

Decision.

Let us be given a circle with center $O$ and diameter $AB$. Draw the tangent $l$ and construct the distances $AD=9\ cm$ and $BC=5\ cm$. Let's draw the radius $OH$ (Fig. 2).

Figure 2.

Since $AD$ and $BC$ are the distances to the tangent, then $AD\bot l$ and $BC\bot l$ and since $OH$ is the radius, then $OH\bot l$, hence $OH |\left|AD\right||BC$. From all this we get that $ABCD$ is a trapezoid, and $OH$ is its midline. By Theorem 1, we get

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The area of ​​the trapezoid. Greetings! In this publication, we will consider this formula. Why is it the way it is and how can you understand it? If there is an understanding, then you do not need to learn it. If you just want to see this formula and what is urgent, then you can immediately scroll down the page))

Now in detail and in order.

A trapezoid is a quadrilateral, two sides of this quadrilateral are parallel, the other two are not. Those that are not parallel are the bases of the trapezium. The other two are called sides.

If the sides are equal, then the trapezoid is called isosceles. If one of the sides is perpendicular to the bases, then such a trapezoid is called rectangular.

In the classical form, the trapezoid is depicted as follows - the larger base is at the bottom, respectively, the smaller one is at the top. But no one forbids depicting it and vice versa. Here are the sketches:

The next important concept.

The median line of a trapezoid is a segment that connects the midpoints of the sides. The median line is parallel to the bases of the trapezoid and is equal to their half-sum.

Now let's delve deeper. Why exactly?

Consider a trapezoid with bases a and b and with the middle line l, and perform some additional constructions: draw straight lines through the bases, and perpendiculars through the ends of the midline until they intersect with the bases:

*Letter designations of vertices and other points are not entered intentionally to avoid unnecessary designations.

Look, triangles 1 and 2 are equal according to the second sign of equality of triangles, triangles 3 and 4 are the same. From the equality of triangles follows the equality of the elements, namely the legs (they are indicated respectively in blue and red).

Now attention! If we mentally “cut off” the blue and red segments from the lower base, then we will have a segment (this is the side of the rectangle) equal to the midline. Further, if we “glue” the cut off blue and red segments to the upper base of the trapezoid, then we will also get a segment (this is also the side of the rectangle) equal to the midline of the trapezoid.

Got it? It turns out that the sum of the bases will be equal to the two medians of the trapezoid:

See another explanation

Let's do the following - build a straight line passing through the lower base of the trapezoid and a straight line that will pass through points A and B:

We get triangles 1 and 2, they are equal in side and adjacent angles (the second sign of equality of triangles). This means that the resulting segment (in the sketch it is marked in blue) is equal to the upper base of the trapezoid.

Now consider a triangle:

*The median line of this trapezoid and the median line of the triangle coincide.

It is known that the triangle is equal to half of the base parallel to it, that is:

Okay, got it. Now about the area of ​​the trapezoid.

Trapezium area formula:

They say: the area of ​​a trapezoid is equal to the product of half the sum of its bases and height.

That is, it turns out that it is equal to the product of the midline and height:

You probably already noticed that this is obvious. Geometrically, this can be expressed as follows: if we mentally cut off triangles 2 and 4 from the trapezoid and put them on triangles 1 and 3, respectively:

Then we get a rectangle in area equal to the area our trapezoid. The area of ​​this rectangle will be equal to the product of the midline and height, that is, we can write:

But the point here is not in writing, of course, but in understanding.

Download (view) the material of the article in *pdf format

That's all. Good luck to you!

Sincerely, Alexander.

A quadrilateral with only two parallel sides is called trapeze.

The parallel sides of a trapezoid are called its grounds, 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 of the trapezium

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

Theorem:

If a line intersecting the midpoint of one side is parallel to the bases of a trapezoid, then it bisects the second lateral side trapezoid.

Theorem:

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

MN midline, AB and CD - bases, AD and BC - 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 line segment connecting the midpoints of the two sides of a triangle is called the midline of the triangle. It is parallel to the third side and its length is 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 given triangle, then it bisects the third side.

AM = MC and BN = NC =>

Applying Triangle and Trapezoid Midline Properties

The division of a segment into a certain number of equal parts.
Task: Divide segment AB into 5 equal parts.
Decision:
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 lines through A 4 , A 3 , A 2 and A 1 that are parallel to A 5 B. They intersect AB at B 4 , B 3 , B 2 and B 1 respectively. 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 get 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 in order 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.