1 Additional construction leading to the theorem on the middle line of the triangle, trapezoids and the properties of the similarity of triangles.
And she equal to half of hypotenuses.
Corollary 1.
Corollary 2.
It seems that 
1 All rectangular triangles with an equal sharp angle are similar. View on trigonometric functions.
Triangles with stroked sides and non-strokes are similar on the equality of two angles. Therefore
This means that these ratios depend only on the acute angle of the rectangular triangle and are essentially determined by it. This is one of the foundations of the appearance of trigonometric functions:
Often the trigonometric function of the angle in such rectangular triangles of visual records of the similarity ratios!
2 Example of additional construction - height, lowered on the hypotenuse. The withdrawal of the Pythagora theorem based on the similarity of triangles.
Power on the hypothenus AB height CH. We have three similar triangles ABC, AHC and CHB. We write expressions for trigonometric functions:
It seems that . Folding, we get the Pythagore's theorem, because:
Another proof of the Pythagoreo Theorem See in Comments on Task 4.
3 An important example of an additional construct is to build an angle equal to one of the corners of the triangle.
We carry out from the vertex of the direct angle of a straight line, which makes up a Ca car angle equal to the CAB angle of a given rectangular triangle ABC. As a result, we obtain an equifiable ACM triangle with angles at the base. But the other triangle, obtained with such a construct, will also be equal, since each angle at the base is equal to (by the property of the corners of the rectangular triangle and on the construction - from the direct angle of "detected" angle). Due to the fact that the triangles of BMC and AMC are eaten with the general side of the MC with the equality MB \u003d MA \u003d MC, i.e. MC - median, spent on the hypotenuz of a rectangular triangle, and she equal to half of hypotenuses.
Corollary 1. The middle of the hypotenuse is the center of the circle described around this triangle, since it turned out that the middle of the hypotenuse is equivalent to the vertices of the rectangular triangle.
Corollary 2. The middle line of the rectangular triangle connecting the middle of the hypotenuse and the middle of the category, parallel to the opposite cathelet and is equal to its half.
Lower in an equally chained triangles of BMC and AMC height MH and MG on the base. Since in an equally chained triangle, the height, lowered to the base, is also median (and bisector), then MH and Mg-rini of a rectangular triangle connecting the middle of the hypotenuse with the middle of the cathets. By construction, they are parallel to opposite customs and their halves, since the triangles are equal to MHC and MGC are equal to (and the MHCG is a rectangle). This result is the basis for proof of the theorem on the midline of an arbitrary triangle and, further, the average line of the trapezoid and the properties of the proportionality of segments that are cut off by parallel straight ones on two cross-line directly.
Tasks
Use of similarity properties -1
Use of basic properties - 2
Using additional constructions 3-4
1 2 3 4
The height, lowered from the vertex of the direct angle of the rectangular triangle is equal to the root of the square of the lengths of the segments to which it divides the hypotenuse.
The decision seems to be obvious if you know the withdrawal of the Pythagoree theorem from the similarity of triangles:
\\ (\\ MathRM (TG) \\ Beta \u003d \\ FRAC (H) (C_1) \u003d \\ FRAC (C_2) (H) \\),
Where \\ (H ^ 2 \u003d C_1C_2 \\).
Find the geometric location (GMT) intersection of the median of all sorts of rectangular triangles, the hypotenuse of which is fixed.
The intersection point of the median of any triangle cuts off from the median one third, counting from the point of its intersection with the corresponding side. In a rectangular triangle, a median conducted from a direct angle is equal to half of the hypotenuse. Therefore, the desired GMT is a circle of radius equal to 1/6 of the length of hypotenuse, with a center in the middle of this (fixed) hypotenuse.
Theme lesson
The middle line of the triangle
Objectives lesson
Consolidate knowledge of schoolchildren about triangles;
Introduce students with such a concept as the middle line of the triangle;
Form knowledge of students about the properties of triangles;
Continue to teach children to apply the properties of figures when solving tasks;
Develop logical thinking, the perishability and attention of students.
Tasks lesson
Form knowledge of schoolchildren about the middle line of triangles;
Check students' knowledge on the topics about triangles;
Check the skill of students to solve problems.
Develop interest in schoolchildren to accurate sciences;
Continue to form the skill of students to express their thoughts and own the mathematical language;
Lesson plan
1. The middle line of the triangle. Basic concepts.
2. The middle line of the triangle, theorems and properties.
3. Repetition of the previously studied material.
4. The main lines of the triangle and their properties.
5. Interesting facts from the field of mathematics.
6. Homework.
The middle line of the triangle
The middle line of the triangle is called such a segment that connects the middle of the two sides of this triangle.
In each triangle there are three middle lines that form another new triangle located inside.
The vertices of the newly formed triangle are on the mids of the sides of this triangle.
Each triangle has the ability to spend three middle lines.
Now let's stop in detail on this topic. Look at the triangle drawing at the top. Before you, the ABC triangle on which the middle line holds. Segments Mn, MP and NP are formed inside this triangle another MNP triangle.
Properties of the middle line of the triangle
Each middle line of the triangle connecting the middle of its sides, has the following properties:
1. The middle line of the triangle is parallel to its third party and is equal to its half.
Thus, we see that the side of the AU is parallel to the Mn, which is two times less than the side of the AU.
2. The middle lines of the triangle divide it into four equal triangles.
If we look at the ABC triangle, we will see that the middle lines Mn, MP and NP were divided into four equal triangles, and as a result, MBN, PMN, NCP and AMP triangles were formed.
3. The middle line of the triangle cuts off from this triangle similar, the area of \u200b\u200bwhich is equal to one fourth source triangle.
For example, in the ABC triangle, the middle line MP cuts off from this triangle, forming the AMP triangle, the area of \u200b\u200bwhich is equal to one fourth ABC triangle.
Triangles
In the previous classes, you have already studied such a geometric shape as a triangle and know what kind of triangles are, what they differ and what properties are possessed.
The triangle refers to the simplest geometric figures that have three sides, three angle and their area is limited to three dots and three sections that pairly connect these points.
So we remembered the definition of a triangle, and now let's repeat everything you know about this figure, answering questions:
4. What types of triangles have you already studied? List them.
5. Give the definitions of each of the types of triangles.
6. What is the triangle area?
7. What is the sum of the angles of this geometric shape?
8. What types of triangles are you known? Name them.
9. What do you know the triangles on the type of equal parties?
10. Give the definition of hypotenuses.
11. How many sharp corners can be in a triangle?
The main lines of the triangle
The main triangle lines include: median, bisector, height and median perpendicular.
Median
The median triangle is called a segment that connects the vertex of the triangle from the middle of the opposite side of this triangle.
Properties Median Triangle
1. It divides the triangle into two other equal in the area;
2. All medians of this figure intersect at one point. This point shares them in relation to two to one, starting the countdown from the top, and is called the center of gravity of the triangle;
3. Medians share this triangle to six areometric.
Bisector
A beam that comes out of the top and passing between the sides of the angle, divides it in half, is called the bisector of this corner.
And if the segment of the corner bisector connects it with a vertex with a point, which lies on the opposite side of the triangle, then it is called a triangle bisector.
Properties of bisector triangle
1. The bisector angle is the geometric location of the points that are equidistant of the sides of this angle.
2. The bisector of the inner corner of the triangle divides the opposite side of the segments, which are proportional to the adjacent sides of the triangle.
3. The center of the circle inscribed in the triangle is the intersection point of the bisector of this figure.
Height
Perpendicular, which is carried out from the top to the figure to a straight line, which is the opposite side of the triangle, is called its height.
Properties of triangle heights
1. The height conducted from the vertex of the direct angle divides the triangle into two similar.
2. If the triangle is acute, then its two heights cut off this triangle to him like.
Municipal perpendicular
The median triangle perpendicular is called direct, which passes through the middle of the segment, which is located perpendicular to this segment.
Properties of middle triangle perpendiculars
1. Any point of the middle perpendicular to the segment is equal to its ends. In this case, the opposite statement will be true.
2. The intersection point of the middle perpendicular, which is carried out to the sides of the triangle, is the center of the circle, which is described near this triangle.
Interesting facts from the field of mathematics
Whether it will learn the news for you that for deciphering the secret correspondence of the Government of Spain, Francois Vieta wanted to send to the fire, because they believed that only the devil could know the cipher, and it could not be forces.
Do you know that the first person who suggested numbers of chairs, ranks and places was Rena Descartes? The theater aristocrats even asked for the King of France to give Descartes for this award, but, alas, the king refused, as he believed that to give a reward of a philosopher - this is lower than his dignity.
Because of the students who could remember the theorem of Pythagora, but could not understand it, this theorem was called an "donkey bridge." This meant that the donkey student, who could not overcome the bridge. In this case, the bridge was considered the theorem of Pythagora.
Fale writers devoted their works not only by mythical heroes, people and animals, but also mathematical symbols. So, for example, the author of the famous "red hat", wrote a fairy tale about the love of the circulation and the line.
Homework
1. Before you are depicted three triangles, give an answer, are the line conducted in the triangles?
2. How many medium lines can be built in one triangle?
3. Dan triangle ABC. Find the ABS triangle sides if its middle lines have such dimensions: of \u003d 5.5 cm, Fn \u003d 8 cm, on \u003d 7 cm.
The average line of the trapezium, and especially its properties, are very often used in geometry to solve problems and evidence of certain theorems.
- This is a quadrangle, who has only 2 parties parallel to each other. Parallel sides are called grounds (in Figure 1 - AD and BC.), two others - side (in the picture AB and CD).
Medium line trapezium - This is a segment connecting the middle of her side sides (in Figure 1 - KL).
Properties of the middle line
Proof of the Middle Line Theorem
ProveIt is equal to the middle line of the trapezium equal to half the grounds and is parallel to these grounds.
Dana trapezium Abcd. with medium line KL. To prove the properties under consideration, it is necessary to spend direct through points. B. and L.. Figure 2 is a straight line BQ.. As well as continue the foundation AD before intersection with a straight BQ..
Consider the resulting triangles LBC. and LQD.:
- By definition of the midline KL dot L. is a middle cut CD. It follows that segments Cl. and LD. equal.
- ∠ BLC = ∠ QLD.Since these corners are vertical.
- ∠ BCL. = ∠ LDQ.since these angles will cover underway with parallel straight lines AD and BC. And Sale CD.
Of these 3 equalities it follows that the previously discussed triangles LBC. and LQD. equal to 1 side and two adjacent angles (see Fig. 3). Hence, ∠ LBC. = ∠ LQD., BC \u003d DQ. and the most important thing - BL \u003d LQ. => KLThe average line of trapezium Abcd.is also the middle line of the triangle ABQ.. According to the property of the middle line of the triangle ABQ. We get.
In solving planimetric problems, in addition to the sides of the corners of the figure, other values \u200b\u200bare often accepted - medians, heights, diagonals, bisector and others. The middle line belongs to their number.
If the original polygon is a trapezium, then what is its middle line? This segment is part of the direct, which crosses the sides of the figure in the middle and is located in parallel to two other parties - the grounds.
How to find a middle line of trapezium through the line of the middle and foundation
If the magnitude of the upper and lower base is known, then the expression will be calculated to calculate the unknown:
a, B - bases, L is the middle line.
How to find an average line of trapezium through the area
If the source data is present in the size of the figure, it is also possible to calculate the length of the line of the trapezium. We use the formula S \u003d (A + B) / 2 * H,
S - Area,
h - height,
A, B - grounds.
But, since L \u003d (a + b) / 2, then S \u003d L * H, which means L \u003d S / H.
How to find the average trapezoid line through the base and corners with it
In the presence of the length of a larger base of the figure, its heights, as well as the known degree of corners with it, the expression for finding the line of the middle of the trapezium will have the following form:
l \u003d a - h * (Ctgα + Ctgβ) / 2, while
L - the desired value
a - greater base
α, β - angles with it,
H is the height of the figure.
If the value of a smaller base is known (with the same other data), the ratio will help to find the middle line:
l \u003d B + H * (CTGα + CTGβ) / 2,
l - the desired value
b is a smaller base
α, β - angles with it,
H is the height of the figure.
Find the middle line of trapezing through the height, diagonal and corners
Consider the situation when in the conditions of the problem there are the values \u200b\u200bof the diagonals of the figure, the angles that they form, crossing each other, as well as the height. Calculate the middle line using expressions:
l \u003d (d1 * d2) / 2h * sinγ or l \u003d (d1 * d2) / 2h * sinφ,
l - line of the middle,
D1, D2 - Diagonal,
φ, γ - angles between them,
H is the height of the figure.
How to find the middle line of the trapezion of an equifiable figure
In case the basic figure - the trapezium is free, the above formulas will have the following form.
- In the presence of values \u200b\u200bof the bases of the trapezing of changes in the expression it will not happen.
l \u003d (a + b) / 2, a, b - base, L is the middle line.
- If the height, base and angles are known, adjacent to it, then:
l \u003d a - H * Ctgα,
L \u003d B + H * Ctgα,
l - line of the middle,
A, B - Bases (B< a),
α - angles with it,
H is the height of the figure.
- If the side of the trapezoid is known and one of the grounds, then you can define the desired value by contacting the expression:
l \u003d a-√ (C * C-H * \u200b\u200bH),
L \u003d B + √ (C * C-H * \u200b\u200bH),
L - line of the middle,
A, B - Bases (B< a),
H is the height of the figure.
- With known height values, diagonals (and they are equal to each other) and the angles formed as a result of their intersection, the interior line can be found as follows:
l \u003d (d * d) / 2h * sinγ or l \u003d (d * d) / 2h * sinφ,
l - line of the middle,
D - diagonal,
φ, γ - angles between them,
H is the height of the figure.
- Square and the height of the figure are known, then:
l \u003d S / H,
S - Area,
H - height.
- If the perpendicular height is unknown, it can be determined by defining a trigonometric function.
h \u003d c * sinα, so
L \u003d S / C * SINα,
L - line of the middle,
S - Area,
C - side,
α-angle at the base.
The middle line of the triangle is an interesting characterizing segment, as it has several properties that allow you to find a simple solution for seemingly complex task. Therefore, consider the basic properties of the middle line and talk about how to find the length of this segment in the triangle.
Triangle and its characterizing segments
The triangle is a figure consisting of three sides and three angles. Depending on the corners, triangles are divided into:
- Otterugal
- Tomber
- Rectangular
Fig. 1. Types of triangles
The main characterizing segments of the triangle are:
- Median - Cut connecting the vertex from the mid-opposite side.
- Bisector - Segment dividing corner in half
- Height - Perpendicular, lowered from the vertex of the triangle on the opposite direction.
Fig. 2. Height, median and bisector in a triangle
For each of the characterizing segments there is its own intersection point. When connecting three intersection points, the median, bisector and heights are the golden cross section of the triangle.
However, there are a number of additional characterizing segments:
- Middle perpendicular - Height restored from the middle of the height. As a rule, the middle perpendicular continues until the intersection on the other side.
- middle line - Cut connecting the middle of the adjacent sides.
- Radius inscribed circle. The inscribed circle is a circle that concerns each side of the triangle.
- The radius of the described circle. The circle described is a circle containing all sides of the triangle.
The adjacent sides of the triangles call the parties that have a total vertex. In geometry there is the concept of opposite sides, i.e. Parties that lie opposite each other and do not have common vertices. But this concept for triangles is not applicable - any pair of parties in the triangle is adjacent.
Medium line property
The properties of the midline are not so much, but they all matter when solving tasks. The fact is that the tasks of finding the length of the midline is not enough, and therefore some of them are able to build a pupil into a stupor with all its simplicity.
Therefore, we give and discuss all the properties of the middle line of the triangle:
- The middle line is equal to half the base. In general, it is more correct to say not half the foundation, but half of the opposite side. Since the sides in the triangle 3, and the base is only one. But in general, the basis can be considered any of the sides of the triangle, so that such a wording is considered permissible. In addition, it is easier to learn. In general, according to this property, the length of the middle line of the triangle is determined.
- The middle line is parallel to the base. With the concept of the foundation here is the same situation as in the past property.
- The middle line cuts off from the triangle a small similar triangle with a similarity ratio of 0.5
- Three medium lines divide a triangle on 4 equal triangles, similar to a large triangle with a similarity ratio of 0.5
Fig. 3. Medium lines in a triangle
The actual formula of the midline length flows from the second property:
$ M \u003d 1 \\ OVER (2) * A $ - where M is the middle line, the side is the opposite midline.
What did we know?
We talked about the secondary characterizing segments, highlighting the average line. They led the properties of medium lines and talked about the features of the formulation of these properties. They described how the formula for the length of the middle line of the triangle is displayed and how the middle line breaks the triangle. All these properties are used when solving triangles.
Test on the topic
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