The article provides the most important theoretical information and necessary solutions specific tasks formulas. Important statements and properties of figures are laid out on the shelves.

Definition and Important Facts

Planimetry is a branch of geometry that deals with objects on a flat two-dimensional surface. Some suitable examples can be identified: square, circle, diamond.

Among other things, it is worth highlighting the point and the straight line. They are the two main concepts of planimetry.

Everything else is built on them, for example:

Axioms and theorems

Let's look at the axioms in more detail. In planimetry this is the most important rules, on which all science works. And not only in it. A-priory, we're talking about about statements that do not require proof.

The axioms that will be discussed below are included in the so-called Euclidean geometry.

• There are two points. You can always draw a single straight line through them.
• If there is a line, then there are points that lie on it and points that do not lie on it.

These 2 statements are usually called axioms of membership, and the following are called axioms of order:

• If there are three points on a straight line, then one of them is necessarily located between the other two.
• A plane is divided by any straight line into two parts. When the ends of a segment lie on one half, then the entire object belongs to it. Otherwise, the original line and the segment have an intersection point.

Axioms of measures:

• Each segment has a length different from zero. If a point splits it into several parts, then their sum will be equal to the total length of the object.
• Each angle has a certain degree measure, which is not equal to zero. If you break it with a beam, then the original angle will be equal to the sum educated.

Parallelism:

• There is a straight line on the plane. Through any point that does not belong to it, only one straight line can be drawn parallel to the given one.

Theorems in planimetry are no longer entirely fundamental statements. They are generally accepted as fact, but each has a proof built on the basic concepts mentioned above. Besides, there are a lot of them. It will be quite difficult to sort everything out, but some of them will be present in the presented material.

The following two are worth familiarizing yourself with early:

• Sum adjacent corners equal to 180 degrees.
• Vertical angles are the same size.

These two theorems can be useful in solving geometric problems associated with n-gons. They are quite simple and intuitive. It's worth remembering them.

Triangles

A triangle is a geometric figure consisting of three segments connected in series. They are classified according to several criteria.

On the sides (the ratios emerge from the names):

At the corners:

• acute-angled;
• rectangular;
• obtuse.

Two angles, regardless of the situation, will always be acute, and the third is determined by the first part of the word. That is, right triangle one of the angles is 90 degrees.

Properties:

• The larger the angle, the larger the opposite side.
• The sum of all angles is 180 degrees.
• The area can be calculated using the formula: S = ½ ⋅ h ⋅ a, where a is the side, h is the height drawn to it.
• You can always inscribe a circle in a triangle or describe it around it.

One of the basic formulas of planimetry is the Pythagorean theorem. It works exclusively for a right triangle and sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs: AB 2 = AC 2 + BC 2.

The hypotenuse means the side opposite the 90° angle, and the legs mean the adjacent ones.

Quadrilaterals

There is an enormous amount of information on this topic. Below are only the most important ones.

Some varieties:

1. Parallelogram - opposite sides equal and pairwise parallel.
2. A rhombus is a parallelogram whose sides have same length.
3. Rectangle - parallelogram with four right angles
4. A square is both a rhombus and a rectangle.
5. Trapezoid - only two opposite sides are parallel.

Properties:

• Suma internal corners equal to 360 degrees.
• The area can always be calculated using the formula: S=√(p-a)(p-b)(p-c)(p-d), where p is half the perimeter, a, b, c, d are the sides of the figure.
• If a circle can be described around a quadrilateral, then I call it convex, if not, non-convex.

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Explanatory note

The offered tickets are intended for oral theoretical transfer annual exam by planimetry 9th grade students secondary school, as well as 10th and 11th grades in order to prepare for the Unified State Exam. The materials offered are fully consistent with the mathematics program and the program for specialized training.

The tickets consist of ten questions reflecting the main directions of the geometry course.

Questions are designed to test mastery conceptual apparatus subject and identifying the level of knowledge of important theoretical facts. Some of them require proof of the material presented, showing knowledge of the basic theoretical principles of the course and the ability to justify them.

These questions are taken from the manuals:

Geometry. Proof problems. Smirnov V.A., Smirnova I.M.

Geometry. Textbook for grades 7-9. Atanasyan, Butuzov, Kadomtsev and others.

Geometry. Textbook for grades 7-11. A.V. Pogorelov.

CRITERIA FOR EVALUATING STUDENTS' ANSWERS

When assessing student responses, you can be guided by the following criteria.

For a complete and correct answer to all questions on the ticket, a score of “5” is given. To get a “3” grade, it is enough to answer eight questions on the ticket.

In all other cases the score is “4”.

Test in planimetry

Option 1

Signs of equality of triangles.

Property midline triangle.

Determining the height of a triangle.

What are the radii of the inscribed and circumscribed circles in a right triangle?

Properties of similar figures.

How is the central angle measured?

Property of chords of a circle.

The center of the circumcircle of a right triangle.

Property of a right triangle that has an acute angle of 30 degrees.

Define the perpendicular bisector.

Option 2

Signs of equality of right triangles.

Determining the median of a triangle.

Pythagorean theorem.

What is the sum of the squares of the diagonals in a parallelogram?

Formula for the area of ​​a regular triangle.

Area of ​​a trapezoid.

Property of inscribed angles.

Property of a circumscribed quadrilateral.

Arc length.

Sine, cosine, tangent of an angle of 30 degrees.

Option 3

Theorem on the sum of the angles of a triangle.

Properties of medians of a triangle.

Determination of the bisector of a triangle.

Cosine theorem.

Formula for the bisector of a triangle.

Area of ​​a parallelogram (3).

Why equal to the angle between two secants intersecting outside the circle.

Property of an inscribed quadrilateral.

Circumference.

Basic properties of chords.

Option 4

Properties of an isosceles triangle.

Property of perpendicular bisectors.

Formula for medians of a triangle.

Theorem of sines.

What are the elements in equilateral triangle(height, radii, area)?

Properties of an isosceles trapezoid.

The property of tangent and secant lines emanating from the same point.

What is the angle between intersecting chords?

Sine, cosine, tangent of an angle of 60 degrees.

Where is the center of the inscribed circle in a triangle?

Option 5

Triangle inequality.

Theorem on the altitudes of a triangle.

Areas of similar triangles.

Formulas for the areas of a triangle (6).

Signs of a parallelogram.

Theorem about the midline of a trapezoid.

Heron's formula for a quadrilateral.

What is the angle between the tangent and the chord drawn from the point of tangency?

Sector area.

Sine, cosine, tangent of an angle of 45 degrees.

Option 6

Determining the midline of a triangle.

Triangle bisector theorem.

Signs of similarity of triangles.

Cosine theorem.

Heron's formula.

Properties of a parallelogram.

Area of ​​a rhombus.

Center of the inscribed and circumscribed circle in a triangle.

Define sine, cosine, tangent and cotangent acute angle right triangle

Average level

Basic axioms of planimetry. Comprehensive guide (2019)

1. Basic concepts of planimetry



Why is everything in pictures and without words? Are words needed? It seems to me that at first they are not very necessary. Actually, mathematicians, of course, know how to describe everything in words, and you can find such descriptions in the following levels of theory, but now let’s continue with pictures.







What else? Oh yes, we need to learn how to measure segments and angles.

Each segment has a length - a number that is assigned to this segment (for some reason...). Length is usually measured ... with a ruler, of course, in centimeters, millimeters, meters and even kilometers.

And now measuring angles. For some reason, angles are usually measured in degrees. Why? There's something for that historical reasons, but we are not dealing with history now. Therefore, we will simply have to take the following agreement for granted.



In a developed angle of degrees.

For brevity they write: . In this case, of course, the magnitude of all other angles can be found if you find out what part of the unfolded angle is given angle. A tool for measuring angles is called a protractor. I think you've seen him more than once in your life.

2. Two Basic Facts About Angles

I. Adjacent angles add up.



This is completely natural, isn't it? After all, adjacent angles together make up a reverse angle!

II. Vertical angles are equal.



Why? And look:

Now what? Well, of course, it follows that. (It is enough, for example, to subtract the second from the first equality. But in fact, you can just look at the picture).

What is the value right angle?

Well, of course, ! After all.

4. Acute and obtuse angle.

That's basically all you need to know to get started. Why didn't we say a word about axioms?

Axioms are the rules of action with the basic objects of planimetry, the very first statements about points and lines. These statements are taken as a basis, not proven.

Why don’t we still formulate and discuss them? You see, the axioms of planimetry, in a sense, simply describe intuitively clear relationships in rather long mathematical language. A clear understanding of the axiomatics is necessary a little later, when you get used to geometric concepts at the level of common sense. Then - welcome to - there's a pretty detailed discussion of the axioms there. In the meantime, try to act like the very ancient Greeks, before the time of Euclid - just solve problems using common sense. I assure you, many tasks will be possible for you!

AVERAGE LEVEL

Imagine that you suddenly find yourself on another planet, or... in a computer game.

In front of you is a set of unknown products, and your task is to prepare as many delicious dishes as possible from this set. What will you need? Of course, rules, instructions - what can be done with certain products. What if you suddenly cook something that is only eaten raw or, conversely, put in a salad something that definitely needs to be boiled or fried? So, without instructions - nowhere!

Okay, but why such an introduction? What does geometry have to do with it? You see, a great many statements about all sorts of figures in geometry are the very many “dishes” that we must learn to cook. But from what? From the basic objects of geometry! But the instructions for their “use” are called with clever words "system of axioms".

So, pay attention!

Basic objects and axioms of planimetry.

Point and line

These are the most important concepts of planimetry. Mathematicians say that these are “indefinable concepts.” How so? But so, you have to start somewhere.

Now the first rules for handling points and lines. These rules of mathematics are called "axioms"- statements that are taken as a basis, from which then everything basic will be deduced (remember that we have a big culinary mission to “cook” geometry?). So, the first series of axioms is called

I. Axioms of belonging.

Please note, this axiom allows you to draw like this:



Like this: there were two points:



And then a straight line was found:

But the other one doesn’t!



If all this seems too obvious to you, then remember that you are on another planet and still didn’t know at all what to do with objects "dot" And "straight".

Ray, segment, angle.

Now we have learned to put points on lines and draw lines through points, so we can already prepare the first simple “dishes” -, line segment,corner.

1) BEAM

Here he is,





2) CUT



Now let's put things in order. The next series of axioms is called:

II. Axioms of order.



Now - the next level. We need instructions on measurement segments and angles. These axioms are called

III. Axioms of measures for segments and angles.

And now it’s completely strange.

IV. Axioms for the existence of a triangle equal to a given one.

Two corollaries of this axiom are clearer:





Well, the last one is legendary parallel axiom!

But first definition:

V. Axiom of parallels.

Well, it's over axioms of planimetry! Are there too many of them? But imagine, they are all needed. For each of them there is a cunning, cunning reasoning, which shows that if this axiom is removed, then the entire edifice of geometry will fall apart! Well, or something will remain that is completely different from what we are used to.

Now, two basic facts about angles!

Adjacent and vertical angles.

The rays forming an angle are called sides of the angle, and their general beginning- top

This is completely simple theorem, Truth?

After all common side adjacent angles simply splits a straight angle into two angles and therefore (ATTENTION: Axiom 3.2 works!) the sum of adjacent angles is equal to the size of the unfolded one, that is.

It's easier to draw than to describe - see the picture.



This is also an easy theorem. Make sure:

Acute and obtuse angle.

BRIEF DESCRIPTION AND BASIC FORMULAS

Axioms of belonging:

• Axiom 1. Whatever the line, there are points that belong to this line and points that do not belong to it.
• Axiom 2. Through any two points you can draw a straight line, and only one.

Axioms of order:

• Axiom 3. Of the three points on a line, one and only one lies between the other two.
• Axiom 4. A straight line lying in a plane divides this plane into two half-planes. If the ends of a segment belong to the same half-plane, then the segment does not intersect the line. If the ends of a segment belong to different half-planes, then the segment intersects a line.

Axioms of measures for segments and angles:

• Axiom 5. Each segment has a certain length, greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
• Axiom 6. Each angle has a certain degree measure greater than zero. The straight angle is equal. The degree measure of an angle is equal to the sum degree measures angles into which it is divided by any ray passing between its sides.

Axioms for the existence of a triangle equal to a given one:

Parallel Axiom:

• Axiom 8. On a plane, through a point not lying on a given line, you can draw at most one straight line parallel to the given one.

Basic facts about angles:

• Theorem. The sum of adjacent angles is equal.

Well, the topic is over. If you are reading these lines, it means you are very cool.

Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!

Now the most important thing.

You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough...

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For successful completion Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.

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People who received a good education, earn much more than those who did not receive it. This is statistics.

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The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.

You won't be asked for theory during the exam.

You will need solve problems against time.

And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.

It's like in sports - you need to repeat it many times to win for sure.

Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!

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This page contains planimetry theorems that a mathematics tutor can use in preparing a capable student for a serious exam: an Olympiad or an exam at Moscow State University (in preparation for the Mechanics and Mathematics of Moscow State University, VMC), for an Olympiad at Higher School Economics, for the Olympics Financial Academy and at MIPT. Knowledge of these facts opens up before the tutor great opportunities on drawing up competition tasks. It is enough to “play out” some of the mentioned theorem on numbers or supplement its elements with simple relationships with others mathematical objects, and you’ll get a pretty decent Olympiad problem. Many properties are present in strong school textbooks as tasks for proof and are not specifically included in the headings and sections of paragraphs. I tried to correct this shortcoming.

Mathematics is an immense subject, and the number of facts that can be identified as theorems is endless. A math tutor cannot physically know and remember everything. Therefore, some tricky relationships between geometric objects each time they are revealed to the teacher anew. Collecting them all on one page at once is physically impossible. Therefore, I will fill the page gradually as I use the theorems in my lessons.

I advise beginning math tutors to be careful in using additional reference materials, since schoolchildren do not know most of these facts.

Mathematics tutor about the properties of geometric shapes

1) The perpendicular bisector to a side of a triangle intersects with the bisector of the angle opposite it on a circle circumscribed about given triangle. This follows from the equality of the arcs into which the perpendicular bisector divides the lower arc, and from the theorem about the inscribed angle in a circle.

2)If a bisector b, a median m and a height h are drawn from one vertex in a triangle, then the bisector will lie between two other segments, and the lengths of all segments obey the double inequality.

3) IN arbitrary triangle the distance from any of its vertices to its orthocenter (the point of intersection of heights) is 2 times more distance from the center of the circle circumscribed around this triangle to the side opposite this vertex. To prove this, you can draw straight lines through the vertices of the triangle parallel to its altitudes. Then use the similarity of the original and resulting triangle.

4) The intersection point of the medians M of any triangle (its center of gravity) together with the orthocenter of the triangle H and the center of the circumcircle (point O) lie on the same prima, and . This follows from the previous property and from the property of the intersection point of medians.

5) The extension of the common chord of two intersecting circles divides the segment of their common tangent into two equal parts. This property is true regardless of the nature of this intersection (that is, the location of the centers of the circles). To prove this, you can use the property of the square of a tangent segment.

6) If a triangle contains a bisector of its angle, then its square is equal to the difference between the products of the sides of the angle and the segments into which the bisector divides the opposite side.

That is, the following equality holds

7) Are you familiar with the situation when the height from the vertex of a right angle is drawn to the hypotenuse? For sure. Did you know that all the resulting triangles are similar? Surely you know. Then you probably don’t know that any corresponding elements of these triangles form an equality that repeats the Pythagorean theorem, that is, for example, , where and are the radii of inscribed circles in small triangles, and is the radius of a circle inscribed in a large triangle.

8)If you come across a random quadruple with all known parties a, b, c and d, then its area can be easily calculated using a formula reminiscent of Heron’s formula:
, where x is the sum of any two opposite corners quadrangle. If a given quadrilateral is inscribed in a circle, then the formula takes the form:
and is called Brahmagupta's formula

9)If your quadrilateral is circumscribed about a circle (that is, the circle is inscribed in it), then the area of ​​the quadrilateral is calculated by the formula