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Formula for finding the midline of a triangle. middle line

The midline of a triangle is a line segment that connects the midpoints of 2 of its sides. Accordingly, each triangle has three median lines. Knowing the quality of the midline, as well as the lengths of the sides of the triangle and its angles, it is possible to find the length of the midline.

You will need

  • Sides of a triangle, angles of a triangle

Instruction

1. Let in a triangle ABC MN be the midline connecting the midpoints of the sides AB (point M) and AC (point N). By property, the midline of the triangle connecting the midpoints of 2 sides is parallel to the third side and equal to its half. This means that the midline MN will be parallel to the side BC and equal to BC/2. Consequently, to determine the length of the midline of the triangle, it is sufficient to know the length of the side of this particular third side.

2. Let us now know the sides whose midpoints are connected by the median line MN, that is, AB and AC, as well as the angle BAC between them. Because MN is the middle line, then AM = AB/2, and AN = AC/2. Then, according to the cosine theorem, objectively: MN^2 = (AM^2)+(AN^2)-2*AM*AN*cos (BAC) = (AB^2/4)+(AC^2/4)-AB*AC*cos(BAC)/2. From here, MN = sqrt((AB^2/4)+(AC^2/4)-AB*AC*cos(BAC)/2).

3. If the sides AB and AC are known, then the midline MN can be found by knowing the angle ABC or ACB. Let, say, the angle ABC be famous. Because, by the property of the midline, MN is parallel to BC, then the angles ABC and AMN are corresponding, and, consequently, ABC = AMN. Then by the law of cosines: AN^2 = AC^2/4 = (AM^2)+(MN^2)-2*AM*MN*cos(AMN). Consequently, the MN side can be found from the quadratic equation (MN^2)-AB*MN*cos(ABC)-(AC^2/4) = 0.

Tip 2: How to find the side of a square triangle

A square triangle is more correctly referred to as a right triangle. The relationships between the sides and angles of this geometric figure are considered in detail in the mathematical discipline of trigonometry.

You will need

  • - paper;
  • - pen;
  • – Bradis tables;
  • - calculator.

Instruction

1. Discover side rectangular triangle with support for the Pythagorean theorem. According to this theorem, the square of the hypotenuse is equal to the sum of the squares of the legs: c2 \u003d a2 + b2, where c is the hypotenuse triangle, a and b are its legs. In order to apply this equation, you need to know the length of any 2 sides of a rectangular triangle .

2. If the conditions specify the dimensions of the legs, find the length of the hypotenuse. To do this, with the support of a calculator, extract the square root of the sum of the legs, each of which is squared in advance.

3. Calculate the length of one of the legs, if the dimensions of the hypotenuse and the other leg are known. Using a calculator, take the square root of the difference between the squared hypotenuse and the driven leg, also squared.

4. If the hypotenuse and one of the acute angles adjacent to it are given in the problem, use the Bradys tables. They give the values ​​of trigonometric functions for a large number of angles. Use the calculator with sine and cosine functions, as well as trigonometry theorems that describe the relationship between the sides and angles of a rectangular triangle .

5. Find the legs using the basic trigonometric functions: a = c*sin ?, b = c*cos ?, where a is the leg opposite the corner?, b is the leg adjacent to the corner?. Similarly, calculate the size of the sides triangle, if the hypotenuse and another acute angle are given: b = c*sin ?, a = c*cos ?, where b is the leg opposite to the angle?, and is the leg adjacent to the angle?

6. In the case when we lead the leg a and the acute angle adjacent to it?, do not forget that in a right triangle the sum of the acute angles is invariably equal to 90 °: ? +? = 90°. Find the value of the angle opposite to the leg a:? = 90° -?. Or use the trigonometric reduction formulas: sin ? = sin (90° -?) = cos?; tg? = tg (90° – ?) = ctg ? = 1/tan?.

7. If we keep the leg a and the acute angle opposite to it?, using Bradis tables, a calculator and trigonometric functions, calculate the hypotenuse using the formula: c=a*sin?, leg: b=a*tg?.

Related videos

scientific work

1. Properties of middle lines

1. Properties of a triangle:

· when all three middle lines are drawn, 4 equal triangles are formed, similar to the original one with a coefficient of 1/2.

the median line is parallel to the base of the triangle and equal to half of it;

· the middle line cuts off a triangle that is similar to the given one, and its area is equal to one quarter of its area.

2. Properties of a quadrilateral:

If in a convex quadrilateral the midline forms equal angles with the diagonals of the quadrilateral, then the diagonals are equal.

· the length of the midline of the quadrilateral is less than half the sum of the other two sides or equal to it if these sides are parallel, and only in this case.

the midpoints of the sides of an arbitrary quadrilateral are the vertices of the parallelogram. Its area is equal to half the area of ​​the quadrilateral, and its center lies at the point of intersection of the midlines. This parallelogram is called the Varignon parallelogram;

· The intersection point of the midlines of the quadrilateral is their common midpoint and bisects the segment connecting the midpoints of the diagonals. In addition, it is the centroid of the vertices of the quadrilateral.

3. Properties of a trapezoid:

the median line is parallel to the bases of the trapezium and is equal to their half-sum;

The midpoints of the sides of an isosceles trapezoid are the vertices of the rhombus.

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Middle line of the triangle

Properties

  • the middle line of the triangle is parallel to the third side and equal to half of it.
  • when all three middle lines are drawn, 4 equal triangles are formed, similar (even homothetic) to the original one with a coefficient of 1/2.
  • the middle line cuts off a triangle that is similar to the given one, and its area is equal to one quarter of the area of ​​the original triangle.

Middle line of the quadrilateral

Middle line of the quadrilateral A line segment that joins the midpoints of opposite sides of a quadrilateral.

Properties

The first line connects 2 opposite sides. The second connects 2 other opposite sides. The third connects the centers of the two diagonals (not all quadrilaterals intersect the centers)

  • If in a convex quadrilateral the midline forms equal angles with the diagonals of the quadrilateral, then the diagonals are congruent.
  • The length of the midline of a quadrilateral is less than or equal to half the sum of the other two sides if these sides are parallel, and only in this case.
  • The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram. Its area is equal to half the area of ​​the quadrilateral, and its center lies at the point of intersection of the midlines. This parallelogram is called the Varignon parallelogram;
  • The intersection point of the midlines of the quadrilateral is their common midpoint and bisects the segment connecting the midpoints of the diagonals. In addition, it is the centroid of the vertices of the quadrilateral.
  • In an arbitrary quadrilateral, the midline vector is equal to half the sum of the base vectors.

Median line of the trapezoid

Median line of the trapezoid- a segment connecting the midpoints of the sides of this trapezoid. The segment connecting the midpoints of the bases of the trapezoid is called the second midline of the trapezoid.

Properties

  • the middle line is parallel to the bases and equal to their half-sum.

see also

Notes


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  • Mean lethal dose
  • Median line of the trapezoid

See what the "Middle Line" is in other dictionaries:

    MIDDLE LINE- (1) a trapezoid is a segment connecting the midpoints of the sides of a trapezoid. The median line of a trapezoid is parallel to its bases and equal to their half sum; (2) a triangle is a segment connecting the midpoints of the two sides of this triangle: the third side in this case ... ... Great Polytechnic Encyclopedia

    MIDDLE LINE- a triangle (trapezoid) is a segment connecting the midpoints of two sides of a triangle (lateral sides of a trapezoid) ... Big Encyclopedic Dictionary

    middle line- 24 center line: An imaginary line passing through the thread profile so that the thickness of the rib is equal to the width of the groove. Source … Dictionary-reference book of terms of normative and technical documentation

    middle line- a triangle (trapezoid), a segment connecting the midpoints of two sides of a triangle (lateral sides of a trapezoid). * * * MIDDLE LINE THE MIDDLE LINE of a triangle (trapezoid), a segment connecting the midpoints of the two sides of the triangle (lateral sides of the trapezoid) ... encyclopedic Dictionary

    middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis 3 mm linija, dalijanti teniso stalo paviršių išilgai pusiau. atitikmenys: engl. center line; midtrack line vok. Mittellini, f rus. middle line … Sporto terminų žodynas

    middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis Linija, dalijanti fechtavimosi kovos takelį į dvi lygias dalis. atitikmenys: engl. center line; midtrack line vok. Mittellini, f rus. middle line … Sporto terminų žodynas

    middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis Linija, dalijanti sporto aikšt(el)ę pusiau. atitikmenys: engl. center line; midtrack line vok. Mittellini, f rus. middle line … Sporto terminų žodynas

    middle line- 1) S. l. triangle, a segment connecting the midpoints of two sides of a triangle (the third side is called the base). S. l. triangle is parallel to the base and equal to half of it; the area of ​​the parts of the triangle into which c divides it. l., ... ... Great Soviet Encyclopedia

    MIDDLE LINE A triangle is a segment that connects the midpoints of two sides of a triangle. The third side of the triangle is called. the base of the triangle. S. l. triangle is parallel to the base and equal to half its length. In any triangle S. l. cuts off from... Mathematical Encyclopedia

    MIDDLE LINE- a triangle (trapezoid), a segment connecting the midpoints of two sides of a triangle (lateral sides of a trapezoid) ... Natural science. encyclopedic Dictionary

Books

  • Ballpoint pen "Jotter Luxe K177 West M" (blue) (1953203) , . Ballpoint pen in a gift box. Letter color: blue. Line: middle. Made in France…

The concept of the midline of a triangle

Let us introduce the concept of the midline of a triangle.

Definition 1

This is a segment connecting the midpoints of the two sides of the triangle (Fig. 1).

Figure 1. The middle line of the triangle

Triangle midline theorem

Theorem 1

The midline of a triangle is parallel to one of its sides and equal to half of it.

Proof.

Let us be given a triangle $ABC$. $MN$ - middle line (as in Figure 2).

Figure 2. Illustration of Theorem 1

Since $\frac(AM)(AB)=\frac(BN)(BC)=\frac(1)(2)$, then the triangles $ABC$ and $MBN$ are similar according to the second triangle similarity criterion. Means

Also, it follows that $\angle A=\angle BMN$ means $MN||AC$.

The theorem has been proven.

Consequences from the triangle midline theorem

Corollary 1: The medians of a triangle intersect at one point and divide the intersection point in a ratio of $2:1$ starting from the vertex.

Proof.

Consider triangle $ABC$, where $(AA)_1,\ (BB)_1,\ (CC)_1$ is its median. Since the medians divide the sides in half. Consider the middle line $A_1B_1$ (Fig. 3).

Figure 3. Illustration of corollary 1

By Theorem 1, $AB||A_1B_1$ and $AB=2A_1B_1$, hence $\angle ABB_1=\angle BB_1A_1,\ \angle BAA_1=\angle AA_1B_1$. Hence the triangles $ABM$ and $A_1B_1M$ are similar according to the first triangle similarity criterion. Then

Similarly, it is proved that

The theorem has been proven.

Consequence 2: The three middle lines of the triangle divide it into 4 triangles similar to the original triangle with similarity coefficient $k=\frac(1)(2)$.

Proof.

Consider triangle $ABC$ with midlines $A_1B_1,\ (\ A)_1C_1,\ B_1C_1$ (Fig. 4)

Figure 4. Illustration of corollary 2

Consider the triangle $A_1B_1C$. Since $A_1B_1$ is the middle line, then

Angle $C$ is the common angle of these triangles. Therefore, triangles $A_1B_1C$ and $ABC$ are similar according to the second similarity criterion for triangles with similarity coefficient $k=\frac(1)(2)$.

Similarly, it is proved that the triangles $A_1C_1B$ and $ABC$, and the triangles $C_1B_1A$ and $ABC$ are similar with similarity coefficient $k=\frac(1)(2)$.

Consider the triangle $A_1B_1C_1$. Since $A_1B_1,\ (\ A)_1C_1,\ B_1C_1$ are the midlines of the triangle, then

Therefore, according to the third similarity criterion for triangles, triangles $A_1B_1C_1$ and $ABC$ are similar with similarity coefficient $k=\frac(1)(2)$.

The theorem has been proven.

Examples of a task on the concept of the middle line of a triangle

Example 1

Given a triangle with sides $16$ cm, $10$ cm and $14$ cm. Find the perimeter of a triangle whose vertices lie at the midpoints of the sides of the given triangle.

Decision.

Since the vertices of the desired triangle lie in the midpoints of the sides of the given triangle, then its sides are the midlines of the original triangle. By Corollary 2, we get that the sides of the desired triangle are $8$ cm, $5$ cm, and $7$ cm.

Answer:$20$ see

Example 2

Triangle $ABC$ is given. The points $N\ and\ M$ are the midpoints of the sides $BC$ and $AB$ respectively (Fig. 5).

Figure 5

Perimeter of triangle $BMN=14$ cm. Find the perimeter of triangle $ABC$.

Decision.

Since $N\ and\ M$ are the midpoints of the sides $BC$ and $AB$, then $MN$ is the midline. Means

By Theorem 1, $AC=2MN$. We get:

Sometimes the topics that are explained at school may not always be clear the first time. This is especially true for a subject such as mathematics. But things become much more complicated when this science begins to be divided into two parts: algebra and geometry.

Each student may have the ability to one of two directions, but especially in the elementary grades, it is important to understand the basics of both algebra and geometry. In geometry, one of the main topics is considered to be the section on triangles.

How to find the midline of a triangle? Let's figure it out.

Basic concepts

To begin with, to figure out how to find the middle line of a triangle, it is important to understand what it is.

There are no restrictions for drawing the midline: the triangle can be any (isosceles, equilateral, right-angled). And all the properties that relate to the middle line will work.

The midline of a triangle is a line segment that connects the midpoints of 2 of its sides. Therefore, any triangle can have 3 such lines.

Properties

To know how to find the middle line of a triangle, let's denote its properties that need to be remembered, otherwise without them it will be impossible to solve problems with the need to designate the length of the middle line, since all the data obtained must be substantiated and argued by theorems, axioms or properties.

Thus, to answer the question: "How to find the midline of the triangle ABC?", It is enough to know one of the sides of the triangle.

Let's give an example

Take a look at the drawing. It represents triangle ABC with midline DE. Note that it is parallel to the base AC in the triangle. Therefore, whatever the value of AC, the middle line DE will be half as large. For example, AC=20 means DE=10, etc.

In such simple ways, you can understand how to find the middle line of a triangle. Remember its basic properties and definition, and then you will never have problems finding its meaning.