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 mathematical speech;
Cultivate interest in the subject.

Lesson Objectives

Arouse interest in knowledge of geometry;
Continue to train students in solving problems;
Arouse cognitive interest in mathematics 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 of the opposite sides of the tetragon?
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 and the other two do not.

The name of such an 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

The trapezoid consists of elements such as the 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.

We can come across this unusual word while watching sports competitions of 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 trapezoids are not only geometric figure or sports acrobatic equipment, but also powerful back muscles, which 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 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.

1. The upper one is adjacent to the neck and is responsible for raising the shoulders upward.
2. The middle one is between the shoulder blades and is involved in lifting the shoulder blades.
3. Lower - in the lower part of the shoulder blades, responsible for lowering the scapular bones in the lower phase of movement.

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The main functions of the trapezius: moving the shoulders in the vertical and horizontal plane, tilting the head back, and lifting the shoulder blades up.

Keeping trapezius in good shape is necessary for any athlete. This will increase your strength in basic exercises, reduce the load on the shoulder joints and ligaments, reduce the curvature of the spine in the cervical region and minimize the risk of damage and injury to the entire shoulder girdle.

• Shrugs are rightfully considered the best exercise for developing the trapezius, but many athletes do them incorrectly. You cannot use your biceps and forearms. Wrist straps help to cope with this very well. The elbows should be almost completely straight throughout the entire approach, then the load will be placed specifically on the trapezius.
• Don't use too much weight. When training the trapezius muscles, it is much more important to work in full amplitude and feel the maximum muscle contraction at the top point, holding it for 1-2 seconds.
• Do not tuck your chin to your chest when doing shrugs. This increases compression of the cervical spine and can lead to injury.
• Trapezius loves pumping. To properly “pump” these muscles with blood, use supersets, combining shrugs of any variation with pulling movements that also include the shoulders in the work, for example, with a close-grip chin row. Another option for increasing intensity is to perform dropsets at the end of each set: reduce the working weight and, without rest, do another set or two with a lighter weight.
• Trapezius is a relatively small muscle group; it is enough to train it once a week. It is optimal to combine it with back or shoulder training. To make the entire shoulder girdle look massive, do not forget to pay enough attention to your deltoids and neck muscles. If you notice that the trapezius has begun to overtake the shoulders in development, which visually makes the figure less wide in the shoulder girdle, just stop doing individual exercises for this muscle group.
• Trapezius training should be short but intense. As a rule, one or two exercises are enough to work this muscle group. Alternate different movements in each workout and perform them in different orders, then you will make progress faster.
• Watch your posture. Often, stooping in the cervical and thoracic spine prevents the trapezius from being fully trained. The athlete simply cannot perform the required movement in full amplitude and feel the muscle contraction.
• Exercise moderately. Overtraining of the trapezius muscles will lead to poor blood circulation in the neck muscles and the entire cervical spine. This is fraught with increased intracranial pressure, headaches and dizziness.
• Performing shrugs does not involve rotating the shoulder joints at the top point. For some reason, many novice athletes are guilty of this. When used with heavy weights, this rotation becomes one of the most damaging movements for your rotator cuff. The correct trajectory of movement involves raising and lowering the weight in one plane; there should be no extraneous movements.

The best exercises for working the trapezius

Now let's look at exercises that will help you achieve maximum results when working your trapezius muscles.

Shrugs with a barbell

- This is the main exercise for the mass of trapezius. Here, mainly their upper part works, since when lifting the barbell is located in front of you. The movement should be amplitude, as if at the top point you are trying to reach your ears with your shoulders. You can work with quite a lot of weight in this movement, so you can better feel the stretch in the muscles at the bottom. If necessary, use wrist straps and an athletic belt.

Use a medium, shoulder-width grip to avoid engaging your shoulders. When lifting, keep the barbell as close to your body as possible and minimize cheating - this method will lead to nothing but increasing the risk of injury from the movement. An alternative option is shrugs in Smith.

- This is an exercise for the upper trapezius. Here it is recommended to use less weight, but do more repetitions, this way you will more easily achieve intense pumping (blood supply to the muscles).

Since in this exercise the hands are turned parallel to each other, the forearms are actively involved in the work. Therefore, concentrate on keeping your arms straight and not bending your elbows. Then you will lift the dumbbells using the force of your trapezius, not your arms. You can also use straps.

To turn dumbbell shrugs into an exercise for your middle and lower trapezius, sit on a bench and lean forward slightly:

This will change the load vector, and you will compress your shoulder blades more strongly at the top. Thereby most of load and will move to the middle and lower parts of the trapezius muscles.

Shrugs in the simulator

For this exercise you will need a lower block and a wide handle. Keeping your back straight, pull your shoulders up and slightly back. The biomechanics of the movement are different from the movements in classic barbell shrugs. By moving your shoulders back, you place more stress on your mid-trapezius and posterior deltoid muscles. This will make your upper back look more massive and lumpy from behind. In addition, the design of the block simulator predetermines a stronger stretch of the muscles at the lowest point, which only increases the efficiency of this exercise.

Shrugs with a barbell behind your back

This is a great exercise for the middle and lower trapezius. It is not entirely suitable for beginners, as it requires a developed muscular frame and good stretching of the shoulder joints.

For convenience, it is recommended to perform this exercise in a Smith machine. At the bottom point, slightly relax all the muscles of the shoulder girdle to lower the barbell as low as possible. But do not forget to keep your lumbar spine perfectly straight. The closer to your back you move the barbell when lifting, the harder your trapezius will work. A more distant position will place more stress on the rear deltoids.

Close-grip barbell row to the chin

- This is a basic exercise that works both the trapezius and shoulders. In this exercise, it is important to take it quite narrowly and keep your elbow above the level of the hand, then you will be able to work in full amplitude and load the entire trapezius muscle area. The wider you go, the more load is placed on the middle deltoids.

Alternative exercises: Smith row to the chin with a close grip, row of two dumbbells to the chin with a close grip, kettlebell row to the chin.

Deadlift

A review of exercises would be incomplete without mentioning. Even its variety is not so important, be it classic, sumo, trap bar deadlift, Romanian deadlift or dumbbell deadlift. In this exercise, there is almost no dynamic load on the muscle group that interests us, but the trapezius carries a powerful static tension throughout the entire approach. Experienced athletes work with serious weights in this exercise, this predetermines the further growth of the trapezius. Therefore, it is powerlifters who, more often than others, can boast of impressive trapezius, even without doing separate exercises for this muscle group.

Also, the trapezoid bears part of the load when performing any horizontal rows for the thickness of the back: or, lower block and others, as well as when using a narrow grip in vertical rows (pull-ups, upper block rows, etc.). Indirectly, the load falls on the trapezius during many exercises for the deltoid muscles, for example, swings with dumbbells while standing, sitting or bending over, with a wide grip, arm abductions in the rear deltoid exercise machine, and others.

Trapezius muscle training program

There is no fundamental difference between trapezius training during periods of gaining muscle mass and cutting. All exercises (except the deadlift) are relatively isolated and can be used at any stage of training.

Training the trapezoid in the gym is a fairly simple task. Find a couple of exercises that work best for you, and steadily improve your performance using various methods of progression. Use the following diagram as a guide:

To effectively train your trapezius at home, a minimal set of equipment is enough: a barbell or dumbbells. An approximate version of a home trapezius workout looks like this:

Many athletes also train their traps on the horizontal bars and parallel bars, performing simulated hanging shrugs. These movements are more static in nature, the amplitude is strictly limited, and it will not be easy to feel the isolated work of the trapezius in them. However, you can try to replace strength training with them if you do not have the opportunity to train with weights.

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 - middle line(segment connecting the midpoints of the sides).

Area of ​​trapezoid

1. Through the half-sum of bases a, b and height h: S = \frac(a + b)(2)\cdot h
2. Through the center line MN and height h: S = MN\cdot h
3. Through the 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.

> Trapezius muscle: the best exercises for training the trapezius at home and in the gym

Trapezius muscle: the best exercises for training the trapezius at home and in the gym

The trapezius muscle, or simply trapezius, is one of the most important parts of the back. It is also one of the largest paired muscles in the body, which athletes often neglect to train. In bodybuilding and fitness, exercises for the trapezius muscles are performed primarily to increase their size and make the neck and shoulders look massive. However, the upper back plays a vital role in every person's life.

Where is the trapezoid, structure and functions

Understanding the structure, functions and effective exercises for the trapezius will not only eliminate negative health consequences, but also create a powerful shoulder girdle, with highlighted trapezoidal “bumps”, which are the calling card of many.

The trapezius is a muscle that forms the outer layer of the upper back. It starts from the neck and reaches almost to the middle of the back. It has the shape of a trapezoid, which is where it got its name. Depending on the level where the trapezius muscle is located, its three parts are distinguished:

1. Ascending (lower) part.
2. Transverse (middle) part.
3. Descending (upper) part.

Although each part has some differences, the functions of the trapezius muscle are common. All three areas work as a single unit. It has a complex structure, since the fibers are located in different directions. In order to pump up the trapezoid, it is worth taking this feature into account, dividing the load into each part.

The main reason why the trapezius back muscle is not given enough attention is the visual aspect. Its upper part is very pronounced, and the middle and lower parts are partially lost against the background of more massive ones. However, the entire array of trapezoids plays a critical role. It serves as a kind of “shield” for one of the most vulnerable and vital parts of the spine, affects posture and is involved in almost all movements of the arms and shoulder girdle.

1. Ascending part of the trapezoid

The ascending part begins from the supraspinous ligaments of the thoracic vertebrae and the spinous processes of the T4-T12 vertebrae. Anatomically, the trapezius muscle in this area is attached to the medial part of the spine of the scapula (via the aponeurosis). The lower part is responsible for pressing the medial edge of the scapula to chest. Also, together with other muscles, the lower part presses the shoulder blade to the body. Due to this, the shoulder girdle is stabilized.

2. Transverse part of the trapezoid

Anatomically, the trapezium in the middle part (transverse) begins from the nuchal ligament, spinous processes and supraspinous ligaments in the region of the C5-T3 vertebrae. Attaches to:

• Akromion.
• The spine of the scapula.
• Acromial end of the clavicle.

The main task is stabilization and adduction of the scapula. It is also the thickest part of the area, so with regular trapezius exercises, it can stand out noticeably under the skin.

3. Descending part of the trapezoid

This part is often called the trapezius muscle of the neck, as it arises from the external occipital protuberance, the cranial part of the nuchal ligament and the medial part of the superior nuchal line. Attachment of trapezius muscle:

• Akromion.
• Lateral part of the clavicle.
• Spinous processes of the cervical vertebrae (from 1 to 4).

The main function is the formation of the external relief of the neck, protection of the spine, fixation of the shoulder blade and shoulder girdle. Also, the upper trapezius muscle actively interacts with other muscles, sharing their functions. She also participates in extension and rotation of the head.

The functions should be considered taking into account the anatomy of the trapezius muscle of the back and the division of the area into three parts:

• Upper – adduction and external rotation of the scapula, elevation and protraction of the shoulder girdle.
• Medium – adduction of the scapula and retraction of the shoulder girdle.
• Lower – adduction, lowering and external rotation of the scapula, lowering the shoulder girdle.

In general, all three parts act as a “coordinator” in many movements of the shoulder girdle, which is why training the trapezius muscle is considered necessary not only in sports, but also for the health of the back and spine.

Features of trapezius training

To pump up the trapezius muscle you do not need to perform a huge number of exercises. It is enough to include 2 to 4 movements in the training program to get stable progression in the area. However, this is the main problem. Due to its “unpopularity” (as well as a lack of understanding of the importance of the muscle), upper back exercises are most often neglected by athletes. This leads to an imbalance in the development of the back muscles, which can negatively affect not only the progression in other exercises, but also the health of the spine and posture.

In addition to visual and aesthetic problems, poor tone can lead to a number of complications, from neck pain to postural distortion with an increased risk of protrusions. Also, one of the most popular consequences of insufficient training of the trapezius is cervical hyperlordosis. Along the chain, it is compensated by the development of lumbar hyperlordosis, which generally leads to serious problems and a deterioration in the quality of life.

The first thing to remember is that exercises for the upper back should not be performed with heavy weight.

Firstly, this will increase the risk of injury, and secondly, the movement will be carried out at the expense of other muscles. As a result, all exercises for the trapezius back muscle will be ineffective for the target area. Only as the strength and volume of the muscle increases, you can increase the weights to pump the trapezius more effectively. Basic problems that can arise from excessive muscle overload(heavy weight, training volumes or incorrect technique):

• Pinched nerves.
• Headache.
• Pain in the neck or shoulders.
• Worsening problems with intervertebral discs.

Basically, all problems occur due to excessive compression. For this reason, trapezius exercises in the gym should be done no more than 1-2 times a week (or no more than 15-18 approaches per week). Also, taking into account the specifics of work on this area multi-repetition approaches should be preferred, 12-20 times per set.

Top 3 best trapezius exercises for home and gym

Almost all movements can be performed with dumbbells or a barbell. Therefore, you don’t have to worry about how to pump up your trapezoid at home. The whole range of possibilities will be available even for home. Moreover, you don’t even need dumbbells for training (although doing the movements is much more convenient with them). To inflate your trapeze at home, you only need a couple of water bottles (5-8 liters) or any weight loaded into bags.

1. Shrugs

The first and most universal movement that must be included in the program to pump up the trapezoid at home or in the gym. Can be performed with dumbbells, barbell or any other weight. It is important to remember the following technical features:

• When performing shrugs, you need to turn your shoulders back (squeezing your shoulder blades together).
• Only the shoulder joints are involved in the exercise; visually the technique is similar to shrugging the shoulders.
• Focus on your traps so that you use them to lift the weight rather than other muscles.

It is optimal to perform shrugs in 3-4 approaches, 12-20 repetitions.

2. Pull to the chin (broach)

The exercise is usually classified as a group for. Although when using a narrow grip, the main load will switch to the trapezius.

To pump up your upper back, include movement on the day, but only in the second half of the session (when the main muscles are tired and will not take the load from the trapezius).

In fact, this is one of the shrug options that became famous thanks to. The big advantage of the exercise is that in addition to the trapezius, they are also actively loaded. This helps improve posture.

Important points when performing the movement:

• One of the most popular mistakes in technique is pulling using the forearms; it needs to be eliminated and controlled.
• You need to carefully select the weight so that the load falls on the trapezius.

Conclusion

To pump up the trapezius muscle, it is enough to perform 2 to 4 exercises per week. This area responds well to load, responding with an increase in strength and mass. The upper part is considered the slowest growing, and it also has the highest visual priority. The middle and lower parts are actively involved in most movements for the back muscles, so they do not need additional training.

Trapezius training in video format

The trapezoid is a metal crossbar 60-65 cm long (in flights - up to 90 cm) and 25-30 mm in diameter. For a more comfortable grip, the crossbar is wrapped with braid, and gymnasts rub their hands with magnesium. Ropes (with a thin cable inside for strength) are attached to the edges of the crossbar, the ends of which are hooked onto the standard. The outside of the ropes is decorated with braid or bandage. The length of the ropes can be adjusted, usually it does not exceed 4 m. Trapezes used in flights are suspended from the mainboard not on ropes, but on thin cables.

Trapezes are single, double and group. Single trapeze is used for female solo performance. It demonstrates exercises performed while hanging and standing. The most typical are hanging on the knees, a “flag” on one hand, balancing on the back, a back plank, rotations around the trapezoid, as well as various “breaks” on the knees, on the heels, on the toes. “Cliffs” can also be more complex, for example: a “break” into the toes from resting on the neck, a “break” to the knees from a position standing on a trapeze in a swing, a “break” into the toes from balancing on the back while swinging.

Many tricks, such as hanging on your heels, hanging on your toes, and “cliffs,” are often performed on a swinging trapeze, as they are more impressive. A demonstration of dynamic exercises is called a performance on a shwung trap.

The traditional finale of a single trapeze routine is lopping (from the English "looping the loop" - a dead loop) - a rotation around the standard. There are several ways of such rotation, for example, standing tall, hooking special protrusions (pins) on the soles of shoes onto the slots of a rotating sleeve put on the standard.
Another way is to rotate on a rigid trapeze, that is, on a trapeze that has solid metal slings on the sides instead of ropes or cables. (This trick was first shown by the Bulgarian artist L. Dobrich in 1905.) Standing on the apparatus, the artist, having reached the greatest swing amplitude, turned over with the trapeze around the standard. Having developed a circular rotation in one direction, he then transferred the rotation in the opposite direction.

Literature:
3.B. Gurevich, On the genres of the Soviet circus, M., 1977.