The median is the segment drawn from the vertex of the triangle to the middle of the opposite side, that is, it divides it in half by the point of intersection. The point at which the median intersects the opposite side from which it comes out is called the base. Through one point, called the point of intersection, passes each median of the triangle. The formula for its length can be expressed in several ways.
Formulas for expressing the length of the median
- Often in problems in geometry, students have to deal with such a segment as the median of a triangle. The formula for its length is expressed in terms of the sides:
where a, b and c are sides. Moreover, c is the side on which the median falls. This is how the simplest formula looks like. Triangle medians are sometimes required for auxiliary calculations. There are other formulas as well.
- If during the calculation two sides of the triangle and a certain angle α located between them are known, then the length of the median of the triangle, lowered to the third side, will be expressed as follows.
Basic properties
- All medians have one common point of intersection O and they are also divided by it in a ratio of two to one, if we count from the top. This point is called the center of gravity of the triangle.
- The median divides the triangle into two others, the areas of which are equal. Such triangles are called equal triangles.
- If you draw all the medians, then the triangle will be divided into 6 equal figures, which will also be triangles.
- If in a triangle all three sides are equal, then in it each of the medians will also be a height and a bisector, that is, perpendicular to the side to which it is drawn, and bisects the angle from which it exits.
- In an isosceles triangle, the median dropped from a vertex that is opposite a side that is not equal to any other will also be the height and the bisector. Medians dropped from other vertices are equal. This is also a necessary and sufficient condition for isosceles.
- If the triangle is the base of a regular pyramid, then the height lowered onto this base is projected to the intersection point of all medians.
- In a right triangle, the median drawn to the longest side is half its length.
- Let O be the point of intersection of the medians of the triangle. The formula below will be true for any point M.
- Another property is the median of a triangle. The formula for the square of its length in terms of the squares of the sides is presented below.
Properties of the sides to which the median is drawn
- If you connect any two points of intersection of the medians with the sides on which they are lowered, then the resulting segment will be the midline of the triangle and be one half from the side of the triangle with which it has no common points.
- The bases of the heights and medians in the triangle, as well as the midpoints of the segments connecting the vertices of the triangle with the point of intersection of the heights, lie on the same circle.
In conclusion, it is logical to say that one of the most important segments is precisely the median of the triangle. Its formula can be used to find the lengths of its other sides.
This page is devoted to a fairly common information resource - the description and calculation of the area of an arbitrary triangle. The difference from other resources is the calculation of the area online, directly in the process of reading the article
Area through height and base
This is the easiest formula to remember. In words, this formula sounds like this - The area of a triangle is half the product of the base of the triangle and its height.
In the case of a right triangle, this expression takes on an even simpler meaning: The area of a right triangle is half the product of two legs
area in terms of sides of a triangle
The area of a triangle expressed in terms of sides has been known for a very long time - it appears in books dating back to the 1st century BC.
This formula can be expressed in different ways, since the formulas for calculating the parameters of a triangle are enough.
But if you try to think in terms of the times before our era, when there were no formulas in the modern representation, there were no variables and signs of the root, then the only axiom on the basis of which Heron created his formula was the Pythagorean theorem. And since in those days, irrational numbers were not yet known, and scientists had a rather skeptical vision of negative ones, integer numbers were used for reflection.
The proof itself will not be here, assuming just that Heron, supplemented an arbitrary Pythagorean triangle to a rectangle, calculated its area, and divided by two.
Area via vertex coordinates
When the coordinates of the vertices of a triangle are known, the area formula can be expressed as follows:
The third-order determinant is easily decomposed, and therefore the calculation of the area, even in manual mode, will not cause any difficulties.
Area through two sides and the angle between them
Square across a side and two corners
A rare task, but even for such initial data, a formula was calculated. An attentive reader immediately sees the "mistake". The heading says that the area is recognized through a side and two angles, that is, through three variables, and all four are present in the formula. How so?
In fact, there is no mistake, knowing one of the basic axioms of the triangle, which says that the sum of the interior angles of a triangle is always (!!) equal to 180 degrees
Therefore, there is nothing difficult, knowing two angles of a triangle, to find out the third.
Area through the medians of a triangle
| Median per side a | |
| Median per side b | |
| Median per side with |
Beautiful formula, isn't it?
containing this section. The point of intersection of the median with the side of the triangle is called base of the median.
- One can also introduce the concept outer median triangle.
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✪ MEDIANS of the bisector and HEIGHTS of the triangle - Grade 7
✪ Median of a triangle. Construction. Properties.
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Properties
Main property
All three medians of a triangle intersect at one point, which is called the centroid or center of gravity of the triangle, and are divided by this point into two parts in a ratio of 2: 1, counting from the top.
Properties of the medians of an isosceles triangle
- In an isosceles triangle, two medians drawn to equal sides of the triangle are equal, and the third median is both the bisector and the height.
- The converse is also true: if two medians in a triangle are equal, then the triangle is isosceles, and the third median is both the bisector and the height of the angle at its apex.
- In an equilateral triangle, all three medians are equal.
Properties of median bases
- Euler's theorem for a circle nine points: the bases of the three heights of an arbitrary triangle, the midpoints of its three sides ( bases of its medians) and the midpoints of the three segments connecting its vertices with the orthocenter , all lie on the same circle (the so-called nine point circle).
- The segment through grounds any two medians of a triangle is its middle line. The midline of a triangle is always parallel to the side of the triangle with which it has no common points.
- Corollary (theorem Thales about parallel segments). The midline of a triangle is half the length of the side of the triangle to which it is parallel.
Other properties
- If triangle versatile (scalene), then its bisector drawn from any vertex lies between the median and height drawn from the same vertex.
- The median divides the triangle into two equal (in area) triangles.
- A triangle is divided by three medians into six triangles of equal area.
- From the segments forming the medians, you can make a triangle, the area of \u200b\u200bwhich will be equal to 3/4 of the entire triangle. The median lengths satisfy the triangle inequality.
- In a right-angled triangle, the median drawn from a right-angled vertex is half the hypotenuse.
- The longer side of the triangle corresponds to the smaller median.
- Straight segment, symmetrical or isogonally conjugate the inner median with respect to the inner bisector is called the symmedian of the triangle. Three simedians pass through one point Lemoine's point.
- Median of an angle of a triangle isotomically conjugated to herself.
Basic ratios
In particular, the sum of the squares of the medians of an arbitrary triangle is 3/4 of the sum of the squares of its sides: m a 2 + m b 2 + m c 2 = 3 4 (a 2 + b 2 + c 2) (\displaystyle m_(a)^(2)+m_(b)^(2)+m_(c)^(2) =(\frac (3)(4))(a^(2)+b^(2)+c^(2))).
- Conversely, one can express the length of an arbitrary side of a triangle in terms of medians:
Lesson 3
The median divides the area of the triangle in half
The two triangles are called equal in size. If they have the same area.
Theorem 1. The median divides the triangle into two triangles of equal area.
Proof:
Let VM be the median of triangle ABC. Let's prove that
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Draw the height BH of the triangle ABC. Then
,
https://pandia.ru/text/78/448/images/image005_99.gif" width="136" height="34 src=">.
https://pandia.ru/text/78/448/images/image007_80.gif" width="217" height="55 src=">.
Q.E.D.
Theorem 2. The medians of a triangle divide it into six triangles of equal area.
From the theorem, in particular, it follows that if the intersection point of the medians of a triangle is connected to all its vertices, then the triangle will be divided into three equal parts.
Task 1 Two medians of a triangle are mutually perpendicular and equal to 3 and 4, respectively. Find the area of the triangle.
Solution.
Let the medians AM and BE be equal to 3 and 4 respectively in the triangle ABC, , K is the point of intersection of the medians.
https://pandia.ru/text/78/448/images/image013_46.gif" width="120" height="47 src=">.
Since triangle ABC is a right triangle with right angle BCA, then 
.
Since medians divide the triangle into 6 equal parts, then .
Answer: 8
Task 2 The medians of the triangle are 6, 8 and 10, find the area of the triangle.
Solution.
Let the medians AM, BE And CD of this triangle are respectively equal to 6, 8 and 10, K is the point of their intersection. Let us postpone on the continuation of the ray BE beyond the point E the segment EF= KE. Connect points C, F and A.
Consider a triangle KAF.
https://pandia.ru/text/78/448/images/image018_31.gif" width="152" height="41 src=">
https://pandia.ru/text/78/448/images/image020_25.gif" width="67" height="19 src=">, since CKAE is a parallelogram (on the basis of a parallelogram: if the diagonals of a quadrilateral are divided by the intersection point in half, up to a given quadrilateral parallelogram), we get 
.
Since https://pandia.ru/text/78/448/images/image023_26.gif" width="125" height="20 src=">, then by the inverse Pythagorean theorem (if the square of one side of the triangle is equal to the sum of the squares two of its other sides, then the triangle is right-angled) the triangle KAF is right-angled and .
Calculate the area of triangle AKF:
https://pandia.ru/text/78/448/images/image026_24.gif" width="104" height="41 src=">.gif" width="104" height="41 src=">.
https://pandia.ru/text/78/448/images/image030_18.gif" width="16 height=41" height="41"> from the area of the triangle itself.
The proof can be seen, for example, in the methodological manual "Support Problems in Planimetry".
Questions for self-examination:
1. What triangles are called equal area?
2. The area of the triangle is equal to S. What is the area of each of the triangles into which it is divided by the median drawn to any side of this triangle?
3. Into how many equal parts does a triangle divide into three medians drawn in it?
4. The area of the triangle is S. The centroid of this triangle is connected to its vertices. What is the area of each of the resulting triangles?
5. The area of a triangle is 48, what is the area of a triangle made up of the medians of this triangle?
6. The area of a triangle made up of the medians of some triangle is 24, what is the area of the triangle?
View answers.
Tasks for independent solution:
1. Two medians of a triangle are mutually perpendicular and equal to 6 and 8, respectively. Find the area of the triangle.
View solution.
2. The medians of the triangle are 3, 4 and 5 find the area of the triangle.
View solution.
3. Triangle ABC, whose sides are 13 cm, 14 cm and 15 cm, is divided into three triangles by segments connecting the point M intersections of the medians of a triangle with the vertices of the triangle. Find the area of a triangle Navy.
View solution.
4. Two sides of a triangle are 10 and 12, and the median drawn to the third is 5. Find the area of the triangle.
View solution.
Properties
- The medians of a triangle intersect at one point, which is called the centroid, and are divided by this point into two parts in a ratio of 2: 1, counting from the top.
- A triangle is divided by three medians into six triangles of equal area.
- The longer side of the triangle corresponds to the smaller median.
- From the vectors that form the medians, you can make a triangle.
- With affine transformations, the median goes to the median.
- The median of a triangle divides it into two equal parts.
Formulas
- The formula for the median through the sides (derived through the Stewart theorem or by completing it to a parallelogram and using the equality in the parallelogram of the sum of the squares of the sides and the sum of the squares of the diagonals):
- Side formula in terms of medians:
If two medians are perpendicular, then the sum of the squares of the sides to which they are dropped is 5 times the square of the third side.
Mnemonic rule
median monkey,
who has a keen eye
jump right in the middle
sides against the top,
where is now.
Notes
see also
Links
Wikimedia Foundation. 2010 .
See what the "Median of a triangle" is in other dictionaries:
Median: The median of a triangle in planimetry, the segment connecting the vertex of the triangle with the midpoint of the opposite side in statistics, the median is the population value that divides the ranked data series in half Median (statistics) ... ... Wikipedia
Median: The median of a triangle in planimetry, the segment connecting the vertex of the triangle to the midpoint of the opposite side Median (statistics) quantile 0.5 Median (trace) the middle line of the trace drawn between the right and left ... Wikipedia
Triangle and its medians. The median of a triangle is a segment inside a triangle that connects the vertex of the triangle with the midpoint of the opposite side, as well as a straight line containing this segment. Contents 1 Properties 2 Formulas ... Wikipedia
A line that connects the vertex of a triangle with the midpoint of its base. A complete dictionary of foreign words that have come into use in the Russian language. Popov M., 1907. median (lat. mediana medium) 1) geol. a segment that connects the vertex of a triangle with ... ... Dictionary of foreign words of the Russian language
Median (from the Latin mediana middle) in geometry, a segment connecting one of the vertices of a triangle with the midpoint of the opposite side. Three M. triangles intersect at one point, which is sometimes called the "center of gravity" of the triangle, so ... Great Soviet Encyclopedia
A triangle is a straight line (or its segment inside a triangle) connecting the vertex of the triangle with the midpoint of the opposite side. Three M. triangles intersect at one point, to paradise is called the center of gravity of the triangle, the centroid, or ... ... Mathematical Encyclopedia
- (from lat. mediana middle) a segment connecting the apex of a triangle with the middle of the opposite side ... Big Encyclopedic Dictionary
MEDIAN, medians, women. (lat. mediana, lit. middle). 1. A straight line drawn from the vertex of a triangle to the middle of the opposite side (mat.). 2. In statistics, for a series of many data, a quantity that has the property that the number of data, ... ... Explanatory Dictionary of Ushakov
MEDIAN, s, female In mathematics: a straight line segment that connects the vertex of a triangle to the midpoint of the opposite side. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
MEDIAN (from lat. mediana middle), a segment connecting the apex of a triangle with the middle of the opposite side ... encyclopedic Dictionary
