The concept of the midline of the trapezoid
To begin with, let's remember which shape 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, the parallel sides are called the bases of the trapezoid, and not parallel - the sides of the trapezoid.
Definition 2
The middle line of a trapezoid is a line segment connecting the midpoints of the sides of the trapezoid.
Centerline theorem for a trapezoid
Now we introduce the theorem on the middle line of a trapezoid and prove it by the vector method.
Theorem 1
The middle line of the trapezoid is parallel to the bases and equal to their half-sum.
Proof.
Let us be given a trapezoid $ ABCD $ with bases $ AD \ and \ BC $. And let $ MN $ - middle line 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 to add vectors. On the one hand, we get that
On the other side
We add the last two equalities, we get
Since $ M $ and $ N $ are the midpoints of the lateral sides of the trapezoid, we will have
We get:
Hence
From the same equality (since $ \ overrightarrow (BC) $ and $ \ overrightarrow (AD) $ are codirectional and, therefore, collinear) we obtain $ MN || AD $.
The theorem is proved.
Examples of tasks on the concept of the middle line 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.
Solution.
Let's denote the middle line 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 diameter of the circle are removed from its tangent by $ 9 $ cm and $ 5 $ cm, respectively. Find the diameter of this circle.
Solution.
Let us be given a circle with center at point $ O $ and diameter $ AB $. Draw the tangent line $ 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 $, therefore, $ OH | \ left | AD \ right || BC $. From all this we get that $ ABCD $ is a trapezoid, and $ OH $ is its middle line. By Theorem 1, we obtain
The concept of the midline of the trapezoid
To begin with, let's remember which shape 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, the parallel sides are called the bases of the trapezoid, and not parallel - the sides of the trapezoid.
Definition 2
The middle line of a trapezoid is a line segment connecting the midpoints of the sides of the trapezoid.
Centerline theorem for a trapezoid
Now we introduce the theorem on the middle line of a trapezoid and prove it by the vector method.
Theorem 1
The middle line of the trapezoid is parallel to the bases and equal to their half-sum.
Proof.
Let us be given a trapezoid $ ABCD $ with bases $ AD \ and \ BC $. And let $ MN $ be the middle line 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 to add vectors. On the one hand, we get that
On the other side
We add the last two equalities, we get
Since $ M $ and $ N $ are the midpoints of the lateral sides of the trapezoid, we will have
We get:
Hence
From the same equality (since $ \ overrightarrow (BC) $ and $ \ overrightarrow (AD) $ are codirectional and, therefore, collinear) we obtain $ MN || AD $.
The theorem is proved.
Examples of tasks on the concept of the middle line 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.
Solution.
Let's denote the middle line 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 diameter of the circle are removed from its tangent by $ 9 $ cm and $ 5 $ cm, respectively. Find the diameter of this circle.
Solution.
Let us be given a circle with center at point $ O $ and diameter $ AB $. Draw the tangent line $ 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 $, therefore, $ OH | \ left | AD \ right || BC $. From all this we get that $ ABCD $ is a trapezoid, and $ OH $ is its middle line. By Theorem 1, we obtain
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§ 49. KEYSTONE.
A quadrilateral in which two opposite sides are parallel and the other two are not parallel is called a trapezoid.
In drawing 252, the quadrilateral ABDC AB || CD, AC || BD. ABDC - trapezoid.
The parallel sides of the trapezoid are called it grounds; AB and CD are the bases of the trapezoid. The other two sides are called lateral sides trapezoid; АС and ВD are the sides of the trapezoid.
If the sides are equal, then the trapezoid is called isosceles.
The ABOM trapezoid is isosceles, since AM = VO (Fig. 253).
A trapezoid in which one of the lateral sides is perpendicular to the base is called rectangular(Fig. 254).
The middle line of a trapezoid is the segment that connects the midpoints of the sides of the trapezoid.
Theorem. The middle line of the trapezoid is parallel to each of its bases and is equal to their half-sum.
Given: OS is the middle line of the trapezoid ABDK, that is, OK = OA and BC = CD (Fig. 255).
It is necessary to prove:
1) OS || КD and OS || AB;
2) 
Proof. Through points A and C we draw a straight line intersecting the extension of the base KD at some point E.
In triangles ABC and DCE:
ВС = СD - by condition;
/
1 = /
2 as vertical,
/
4 = /
3, as internal crosswise lying with parallel AB and KE and secant BD. Hence, /\
ABC = /\
DCE.
Hence AC = CE, i.e. OS is the middle line of the KAE triangle. Therefore (§ 48):
1) OS || KE and, therefore, OS || КD and OS || AB;
2) 
, but DE = AB (from the equality of triangles ABC and DCE), therefore the segment DE can be replaced by the segment AB equal to it. Then we get:
The theorem is proved.
Exercises.
1. Prove that the sum inner corners trapezoid adjacent to each side is equal to 2 d.
2. Prove that the angles at the base of an isosceles trapezoid are equal.
3. Prove that if the angles at the base of a trapezoid are equal, then this trapezoid is isosceles.
4. Prove that the diagonals of an isosceles trapezoid are equal.
5. Prove that if the diagonals of a trapezoid are equal, then this trapezoid is isosceles.
6. Prove that the perimeter of the figure formed by the segments connecting the midpoints of the sides of the quadrilateral is equal to the sum of the diagonals of this quadrilateral.
7. Prove that a straight line passing through the middle of one of the lateral sides of the trapezoid parallel to its bases divides the other lateral side of the trapezoid in half.
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you leave a request on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and report unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notifications and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, competition, or similar promotional event, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose information received from you to third parties.
Exceptions:
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- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to an appropriate third party - the legal successor.
Protection of personal information
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Respect for your privacy at the company level
In order to make sure that your personal information is safe, we bring the rules of confidentiality and security to our employees, and strictly monitor the implementation of confidentiality measures.