The straight line segment connecting the midpoints of the lateral sides of the trapezoid is called the midline of the trapezoid. About how to find midline trapezium and how it relates to other elements of this figure, we will describe below.
Centerline theorem
Let's draw a trapezoid in which AD - larger base, BC - smaller base, EF - middle line. Let's extend the base AD beyond point D. Draw a line BF and continue it until it intersects with the continuation of the base AD at point O. Consider the triangles ∆BCF and ∆DFO. Angles ∟BCF = ∟DFO as vertical. CF = DF, ∟BCF = ∟FDО, because VS // JSC. Therefore, triangles ∆BCF = ∆DFO. Hence the sides BF = FO.
Now consider ∆ABO and ∆EBF. ∟ABO is common to both triangles. BE/AB = ½ by condition, BF/BO = ½, since ∆BCF = ∆DFO. Therefore, triangles ABO and EFB are similar. Hence the ratio of the parties EF/AO = ½, as well as the ratio of the other parties.
We find EF = ½ AO. The drawing shows that AO = AD + DO. DO = BC as sides equal triangles, which means AO = AD + BC. Hence EF = ½ AO = ½ (AD + BC). Those. the length of the midline of a trapezoid is equal to half the sum of the bases.
Is the midline of a trapezoid always equal to half the sum of the bases?
Suppose there is such special case, when EF ≠ ½ (AD + BC). Then BC ≠ DO, therefore, ∆BCF ≠ ∆DCF. But this is impossible, since they have two equal angles and sides between them. Therefore, the theorem is true under all conditions.
Midline problem
Suppose, in our trapezoid ABCD AD // BC, ∟A = 90°, ∟C = 135°, AB = 2 cm, diagonal AC is perpendicular to the side. Find the midline of the trapezoid EF.
If ∟A = 90°, then ∟B = 90°, which means ∆ABC is rectangular.
∟BCA = ∟BCD - ∟ACD. ∟ACD = 90° by convention, therefore, ∟BCA = ∟BCD - ∟ACD = 135° - 90° = 45°. 
If in a right triangle ∆ABC one angle is equal to 45°, then the legs in it are equal: AB = BC = 2 cm.
Hypotenuse AC = √(AB² + BC²) = √8 cm.
Let's consider ∆ACD. ∟ACD = 90° according to the condition. ∟CAD = ∟BCA = 45° as the angles formed by the transversal of the parallel bases of the trapezoid. Therefore, legs AC = CD = √8.
Hypotenuse AD = √(AC² + CD²) = √(8 + 8) = √16 = 4 cm.
Midline of trapezoid EF = ½(AD + BC) = ½(2 + 4) = 3 cm.
In this article, another selection of problems with trapezoid has been made for you. The conditions are somehow related to its midline. Task types taken from open bank typical tasks. If you wish, you can refresh your theoretical knowledge. The blog has already discussed tasks whose conditions are related to, as well as. Briefly about the middle line:
The midline of the trapezoid connects the midpoints of the lateral sides. It is parallel to the bases and equal to their half-sum.
Before solving problems, let's look at a theoretical example.
Given a trapezoid ABCD. Diagonal AC intersecting with the middle line forms point K, diagonal BD forms point L. Prove that segment KL equal to half base differences.
Let's first note the fact that the midline of a trapezoid bisects any segment whose ends lie on its bases. This conclusion suggests itself. Imagine a segment connecting two points of the bases, it will split this trapezoid to the other two. It turns out that the segment parallel to the bases trapezoid and passing through the middle of the side on the other side will pass through its middle.
This is also based on Thales' theorem:
If on one of two straight lines we plot several equal segments and through their ends draw parallel lines intersecting the second line, then they will cut off equal segments on the second line.
That is, in in this case K is the middle of AC and L is the middle of BD. Therefore EK is the middle line triangle ABC, LF is the midline of triangle DCB. According to the property of the midline of a triangle:
We can now express the segment KL in terms of bases:
Proven!
This example is given for a reason. In tasks for independent decision there is just such a task. Only it doesn’t say that the segment connecting the midpoints of the diagonals lies on the midline. Let's consider the tasks:
27819. Find the midline of the trapezoid if its bases are 30 and 16.
We calculate using the formula:
27820. The midline of the trapezoid is 28 and the smaller base is 18. Find the larger base of the trapezoid.
Let's express the larger base:
Thus:
27836. Perpendicular dropped from the vertex obtuse angle on a larger basis isosceles trapezoid, divides it into parts having lengths 10 and 4. Find the midline of this trapezoid.
In order to find the middle line you need to know the bases. The base AB is easy to find: 10+4=14. Let's find DC.
Let's construct the second perpendicular DF:
The segments AF, FE and EB will be equal to 4, 6 and 4 respectively. Why?
In an isosceles trapezoid, perpendiculars lowered to the larger base divide it into three segments. Two of them, which are cut off legs right triangles, are equal to each other. The third segment is equal to the smaller base, since when constructing the indicated heights a rectangle is formed, and in the rectangle opposing sides are equal. In this task:
Thus DC=6. We calculate:
27839. The bases of the trapezoid are in the ratio 2:3, and the midline is 5. Find the smaller base.
Let's introduce the proportionality coefficient x. Then AB=3x, DC=2x. We can write:
Therefore, the smaller base is 2∙2=4.
27840. The perimeter of an isosceles trapezoid is 80, its midline is equal to the lateral side. Find side trapezoids.
Based on the condition, we can write:
If we denote the middle line through the value x, we get:
The second equation can already be written as:
27841. The midline of the trapezoid is 7, and one of its bases is 4 greater than the other. Find the larger base of the trapezoid.
Let us denote the smaller base (DC) as x, then the larger one (AB) will be equal to x+4. We can write it down
We found that the smaller base is early five, which means the larger one is equal to 9.
27842. The midline of the trapezoid is 12. One of the diagonals divides it into two segments, the difference of which is 2. Find the larger base of the trapezoid.
We can easily find the larger base of the trapezoid if we calculate the segment EO. It is the midline in triangle ADB, and AB=2∙EO.
What do we have? It is said that the middle line is equal to 12 and the difference between the segments EO and ОF is equal to 2. We can write two equations and solve the system:
It is clear that in this case you can select a pair of numbers without calculations, these are 5 and 7. But, nevertheless, let’s solve the system:
So EO=12–5=7. Thus, the larger base is equal to AB=2∙EO=14.
27844. In an isosceles trapezoid, the diagonals are perpendicular. The height of the trapezoid is 12. Find its midline.
Let us immediately note that the height drawn through the intersection point of the diagonals in an isosceles trapezoid lies on the axis of symmetry and divides the trapezoid into two equal ones rectangular trapezoids, that is, the bases of this height are divided in half.
It would seem that to calculate the middle line we must find reasons. Here a small dead end arises... How, knowing the height, in this case, calculate the bases? No way! There are many such trapezoids with a fixed height and diagonals intersecting at an angle of 90 degrees. What should I do?
Look at the formula for the midline of a trapezoid. After all, we do not need to know the reasons themselves; it is enough to know their sum (or half-sum). We can do this.
Since the diagonals intersect at right angles, isosceles right triangles are formed with height EF:
From the above it follows that FO=DF=FC, and OE=AE=EB. Now let’s write down what the height is equal to, expressed through the segments DF and AE:
So the middle line is 12.
*In general, this is a task, as you understand, for mental counting. But I'm sure the presented detailed explanation necessary. And so... If you look at the figure (provided that the angle between the diagonals is observed during construction), the equality FO=DF=FC, and OE=AE=EB immediately catches your eye.
The prototypes also include types of tasks with trapezoids. It is built on a sheet of paper in a square and you need to find the middle line; the side of the cell is usually equal to 1, but it can be a different value.
27848. Find the midline of the trapezoid ABCD, if the sides of square cells are equal to 1.
It's simple, calculate the bases by cells and use the formula: (2+4)/2=3
If the bases are built at an angle to the cell grid, then there are two ways. For example!
The midline of the trapezoid is equal to half the sum grounds. It connects the midpoints of the sides of the trapezoid and is always parallel to the bases.
If the bases of a trapezoid are equal to a and b, then the middle line m is equal to m=(a+b)/2.
If the area of the trapezoid is known, then the middle line can be found and in another way, dividing the area of the trapezoid S by the height of the trapezoid h:
That is, midline of trapezoid m=S/h
There are many ways to find the length of the midline of a trapezoid. The choice of method depends on the initial data.
Here formulas for the length of the midline of a trapezoid:
To find the midline of a trapezoid, you can use one of five formulas (I won’t write them out, since they are already in other answers), but this is only in cases where the values of the initial data we need are known.
In practice, we have to solve many problems when there is insufficient data and right size still need to find it.
There are such options here
a step-by-step solution to bring everything under the formula;
using other formulas, compose and solve the necessary equations.
finding the length of the middle of a trapezoid using the formula we need with the help of other knowledge about geometry and using algebraic equations:
We have an isosceles trapezoid, its diagonals intersect at right angles, its height is 9 cm.
We make a drawing and see that this problem cannot be solved head-on (there is not enough data)
Therefore, we will simplify a little and draw the height through the point of intersection of the diagonals.
This is the first important step which leads to a quick solution.
let's designate the height with two unknowns, we will see the ones we need isosceles triangles with the parties X And at
and we can easily find it sum of grounds trapezoids
it is equal 2х+2у
And only now we can apply the formula where
and it is equal x+y and according to the conditions of the problem, this is the length of the height equal to 9 cm.
And now we have derived several moments for an isosceles trapezoid, the diagonals of which intersect at right angles
in such trapezoids
the middle line is always equal to the height
area is always equal to the square of the height.
The midline of a trapezoid is a segment that connects the midpoints of the sides of the trapezoid.
The midline of any trapezoid is easy to find if you use the formula:
m = (a + b)/2
m is the length of the midline of the trapezoid;
a, b lengths of the bases of the trapezoid.
So, the length of the midline of a trapezoid is equal to half the sum of the lengths of the bases.
The basic formula for the formula for the midline of a trapezoid: the length of the midline of a trapezoid is equal to half the sum of the bases a and b: MN=(a+b)2. The proof of this formula is the formula for the midline of a triangle. Any trapezoid can be represented after drawing from the ends a smaller base of height to a larger base. The 2 resulting triangles and a rectangle are considered. After this, the formula for the midline of the trapezoid is easily proven.
To find the midline of the trapezoid we need to know the values of the bases.
After we found these values, or maybe they were known to us, we add up these numbers and simply divide them in half.
This is what will happen midline of trapezoid.
As far as I remember my school geometry lessons, in order to find the length of the midline of a trapezoid, you need to add the lengths of the bases and divide by two. Thus, the length of the midline of the trapezoid is equal to half the sum of the bases.