How to find the largest median of a triangle. Median

The median and height of a triangle is one of the most fascinating and interesting topics geometry. The term "median" means a line or segment that connects the vertex of a triangle with its opposite side. In other words, the median is a line that runs from the middle of one side of a triangle to the opposite vertex of the same triangle. Since a triangle has only three vertices and three sides, there can only be three medians.

Triangle median properties

1. All medians of a triangle intersect at one point and are separated by this point in a ratio of 2:1, counting from the top. Thus, if you draw all three medians in a triangle, then the point of their intersection will divide them into two parts. The part that is closer to the top will be 2/3 of the entire line, and the part that is closer to the side of the triangle will be 1/3 of the line. The medians intersect at one point.
2. Three medians drawn in one triangle divide this triangle into 6 small triangles, whose area will be equal.
3. The larger the side of the triangle from which the median comes, the smaller this median. Conversely, the most short side has the longest median.
4. Median in right triangle has a number of its own characteristics. For example, if a circle is described around such a triangle, which will pass through all the vertices, then the median right angle, drawn to the hypotenuse, will become the radius of the circumscribed circle (that is, its length will be the distance from any point on the circle to its center).

Triangle median length equation

The median formula comes from Stewart's theorem and states that the median is Square root from the ratio of the squares of the sum of the sides of the triangle that form the vertex, minus the square of the side to which the median is drawn to four. In other words, to find out the length of the median, you need to square the lengths of each side of the triangle, and then write it as a fraction, the numerator of which will be the sum of the squares of the sides that form the angle from which the median comes, minus the square of the third side. The denominator here is the number 4. Then, from this fraction, you need to extract the square root, and then we get the length of the median.

Point of intersection of the medians of a triangle

As we wrote above, all medians of one triangle intersect at one point. This point is called the center of the triangle. It divides each median into two parts, the length of which is related as 2:1. The center of the triangle is also the center of the circle circumscribed around it. And others geometric figures have their own centres.

The coordinates of the point of intersection of the medians of the triangle

To find the coordinates of the intersection of the medians of one triangle, we use the property of the centroid, according to which it divides each median into 2:1 segments. We denote the vertices as A(x 1 ;y 1), B(x 2 ;y 2), C(x 3 ;y 3),

and calculate the coordinates of the center of the triangle by the formula: x 0 = (x 1 + x 2 + x 3) / 3; y 0 \u003d (y 1 + y 2 + y 3) / 3.

Area of ​​a triangle in terms of the median

All medians of one triangle divide this triangle by 6 equal triangles, and the center of the triangle divides each median by a ratio of 2:1. Therefore, if the parameters of each median are known, it is possible to calculate the area of ​​the triangle through the area of ​​one of the small triangles, and then increase this figure by 6 times.

Instruction

To withdraw formula for medians in an arbitrary , it is necessary to turn to the corollary of the cosine theorem for a parallelogram obtained by completing triangle. The formula can be proved on this, it is very convenient when solving if all the lengths of the sides are known or they can be easily found from other initial data of the problem.

In fact, the cosine theorem is a generalization of the Pythagorean theorem. It sounds like this: for a two-dimensional triangle with side lengths a, b and c and angle α opposite a, the following equality holds: a² = b² + c² - 2 b c cos α.

The generalizing corollary from the cosine theorem defines one of the most important properties quadrilateral: the sum of the squares of the diagonals is equal to the sum of the squares of all its sides: d1² + d2² = a² + b² + c² + d².

Complete the triangle to parallelogram ABCD by adding lines parallel to a and c. thus with sides a and c and diagonal b. The most convenient way to build is as follows: on the straight line to which the median belongs, a segment MD of the same length, connect its vertex to the vertices of the remaining A and C.

According to the property of a parallelogram, the diagonals are divided by the intersection point into equal parts. Apply the corollary of the cosine theorem, according to which the sum of the squares of the diagonals of a parallelogram is equal to the sum of twice the squares of its sides: BK² + AC² = 2 AB² + 2 BC².

Since BK = 2 BM and BM is the median of m, then: (2 m) ² + b² = 2 c² + 2 a², hence: m = 1/2 √(2 c² + 2 a² - b²).

you brought out formula one of triangle for side b: mb = m. Similarly, there are medians its other two sides: ma = 1/2 √(2 c² + 2 b² - a²); mc = 1/2 √(2 a² + 2 b² - c²).

Sources:

• median formula
• Formulas for the median of a triangle [video]

median triangle is called a segment connecting any vertex triangle with the middle of the opposite side. Three medians intersect at one point always inside triangle. This point divides each median in a ratio of 2:1.

Instruction

The problem of finding the median can be solved by additional constructions triangle to a parallelogram and through the theorem on the diagonals of a parallelogram. Let's extend the sides triangle and median, building them up to a parallelogram. So the median triangle will be half the diagonal of the resulting parallelogram, two sides triangle- its side (a, b), and the third side triangle, to which the median was drawn, is the second diagonal of the resulting parallelogram. According to the theorem, the sum of the squares of a parallelogram is equal to twice the sum of the squares of its sides.
2*(a^2 + b^2) = d1^2 + d2^2,
where
d1, d2 - diagonals of the resulting parallelogram;
from here:
d1 = 0.5*v(2*(a^2 + b^2) - d2^2)

The median is the line segment that connects the vertex triangle and the middle of the opposite side. Knowing the lengths of all three sides triangle, you can find its medians. In particular cases of isosceles and equilateral triangle, obviously, it is enough to know, respectively, two (not equal to each other) and one side triangle.

You will need

• Ruler

Instruction

Consider general case triangle ABC with unequal friend parties. The length of the median AE of this triangle can be calculated using the formula: AE = sqrt(2*(AB^2)+2*(AC^2)-(BC^2))/2. The rest of the medians are exactly the same. This is derived through the Stewart theorem, or through completion triangle to a parallelogram.

If ABC is isosceles and AB = AC, then the median AE will be both this triangle. Therefore, triangle BEA will be a right triangle. By the Pythagorean theorem, AE = sqrt((AB^2)-(BC^2)/4). From the total length of the median triangle, for medians BO and СP it is true: BO = CP = sqrt(2*(BC^2)+(AB^2))/2.

Sources:

• Medians and nonsectors of a triangle

The median is the line segment that connects the vertex of a triangle and the midpoint of the opposite side. Knowing the lengths of all three sides of a triangle, you can find it medians. In particular cases of isosceles and equilateral triangle, obviously, it is enough to know, respectively, two (not equal to each other) and one side of the triangle. The median can also be found from other data.

You will need

• The lengths of the sides of the triangle, the angles between the sides of the triangle

Instruction

Consider the most general case triangle ABC with three unequal sides. Length medians The AE of this triangle can be calculated using the formula: AE = sqrt(2*(AB^2)+2*(AC^2)-(BC^2))/2. Rest medians are exactly the same. This is derived through Stewart's theorem, or through the completion of a triangle to a parallelogram.

If ABC is isosceles and AB = AC, then AE will be at the same time this triangle. Therefore, triangle BEA will be a right triangle. By the Pythagorean theorem, AE = sqrt((AB^2)-(BC^2)/4). Of the total length medians triangle, for BO and CP it is true: BO = CP = sqrt(2*(BC^2)+(AB^2))/2.

The median of a triangle can also be found from other data. For example, if the lengths of two sides are given, a median is drawn to one of them, for example, the lengths of the sides AB and BC, as well as the angle x between them. Then the length medians can be found through the cosine theorem: AE = sqrt((AB^2+(BC^2)/4)-AB*BC*cos(x)).

Sources:

• Medians and bisectors of a triangle
• how to find the length of the median

1. What is the median?

It's very simple!

Take the triangle

Mark the middle on one of its sides.

And connect with the opposite top!

The resulting line and is the median.

2. Properties of the median.

What are the good properties of the median?

1) Let's imagine that the triangle - rectangular. There are those, right?

Why??? What's with the right angle?

Let's look carefully. Only not on a triangle, but on ... a rectangle. Why, you ask?

But you walk on the Earth - do you see that it is round? No, of course, for this you need to look at the Earth from space. So we look at our right-angled triangle "from space".

Let's draw a diagonal:

Do you remember that the diagonals of a rectangle equal and share intersection point in half? (If you do not remember, look at the topic)

So half of the second diagonal is ours median. The diagonals are equal, their halves, of course, too. Here we get

We will not prove this statement, but in order to believe in it, think for yourself: is there any other parallelogram with equal diagonals, except for a rectangle? Of course not! Well, that means that the median can be equal to half of the side only in a right triangle.

Let's see how this property helps solve problems.

Here, task:
To the sides; . From the top held median. Find if.

Hooray! You can apply the Pythagorean theorem! See how great it is? If we didn't know that median equal to half a side

We apply the Pythagorean theorem:

2) And now let us have not one, but whole three medians! How do they behave?

Remember very important fact:

Complicated? Look at the picture:

The medians and intersect at one point.

And .... (we prove it in , but for now Remember!):

• - twice as much as;
• - twice as much as;
• - double that.

Not tired yet? Enough strength for the next example? Now we will apply everything we talked about!

Task: In a triangle, medians and are drawn, which intersect at a point. Find if

We find by the Pythagorean theorem:

And now we apply the knowledge about the point of intersection of medians.

Let's mark it. cut, a. If not everything is clear - look at the picture.

We have already found that.

Means, ; .

In the problem we are asked about a segment.

in our notation.

Answer: .

Liked? Now try to apply knowledge about the median yourself!

MEDIAN. MIDDLE LEVEL

1. The median bisects the side.

And all? Or maybe she even divides something in half? Imagine that it is!

2. Theorem: The median bisects the area.

Why? And let's remember the most simple form area of ​​a triangle.

And we apply this formula twice!

Look, the median divided into two triangles: and. But! They have the same height! Only at this height falls to the side, and at - for the continuation of the side. Surprisingly, it also happens like this: the triangles are different, but the height is the same. And so, now we apply the formula twice.

What would that mean? Look at the picture. In fact, there are two statements in this theorem. Did you notice it?

First statement: medians intersect at one point.

Second statement: the intersection point of the median is divided in relation, counting from the top.

Let's try to unravel the secret of this theorem:

Let's connect the dots and. What happened?

And now let's draw another middle line: mark the middle - put a point, mark the middle - put a point.

Now - the middle line. I.e

1. parallel;

Did you notice any coincidences? Both and are parallel. And, and.

What follows from this?

1. parallel;

Of course, only a parallelogram!

So - parallelogram. So what? And let's remember the properties of a parallelogram. For example, what do you know about the diagonals of a parallelogram? That's right, they divide the intersection point in half.

Let's look at the picture again.

That is - the median is divided by points and into three equal parts. And just the same.

This means that both medians separated by a point precisely in relation, that is, and.

What will happen to the third median? Let's go back to the beginning. Oh God?! No, now everything will be much shorter. Let's drop the median and draw the medians and.

Now imagine that we have carried out exactly the same reasoning as for the medians and. What then?

It turns out that the median will divide the median in exactly the same way: in relation, counting from the point.

But how many points can there be on a segment that divide it in relation, counting from a point?

Of course, only one! And we have already seen it - this is the point.

What happened in the end?

The median exactly passed through! All three medians passed through it. And everyone was divided in relation, counting from the top.

So we solved (proved) the theorem. The answer turned out to be a parallelogram sitting inside a triangle.

4. The formula for the length of the median

How to find the length of the median if the sides are known? Are you sure you need it? Let's open terrible secret: This formula is not very useful. But still, we will write it, but we will not prove it (if you are interested in the proof, see the next level).

How would one understand why this happens?

Let's look carefully. Only not on a triangle, but on a rectangle.

So let's look at a rectangle.

Have you noticed that our triangle is exactly half of this rectangle?

Let's draw a diagonal

Do you remember that the diagonals of a rectangle are equal and bisect the intersection point? (If you do not remember, look at the topic)
But one of the diagonals is our hypotenuse! So the point of intersection of the diagonals is the midpoint of the hypotenuse. She was called by us.

So half of the second diagonal is our median. The diagonals are equal, their halves, of course, too. Here we get

Moreover, this happens only in a right triangle!

We will not prove this statement, but in order to believe in it, think for yourself: is there any other parallelogram with equal diagonals, except for a rectangle? Of course not! Well, that means that the median can be equal to half of the side only in a right triangle. Let's see how this property helps solve problems.

Here is the task:

To the sides; . The median is drawn from the top. Find if.

Hooray! You can apply the Pythagorean theorem! See how great it is? If we didn't know that the median is half the side only in a right triangle, we could not solve this problem in any way. And now we can!

We apply the Pythagorean theorem:

MEDIAN. BRIEFLY ABOUT THE MAIN

1. The median bisects the side.

2. Theorem: The median bisects the area

4. The formula for the length of the median

Inverse theorem: if the median is equal to half of the side, then the triangle is right-angled and this median is drawn to the hypotenuse.

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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 most simple formula. 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 the intersections of O and it are divided 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.
• AT 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. It is also necessary and sufficient condition isosceles.
• If the triangle is the base correct pyramid, then the height dropped to the given 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 we connect any two points of intersection of the medians with the sides on which they are lowered, then the resulting segment will be middle line triangle and be one half of 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.