A trapezoid is a convex quadrilateral in which two opposite sides are parallel and the other two are non-parallel. If all opposite sides of a quadrilateral are parallel in pairs, then it is a parallelogram.

You will need

• - all sides of the trapezoid (AB, BC, CD, DA).

Instructions

• Non-parallel sides trapezoids are called laterals, and parallel ones are called bases. The line between the bases, perpendicular to them - height trapezoids. If the sides trapezoids are equal, then it is called isosceles. First let's look at the solution for trapezoids, which is not isosceles.
• Draw line segment BE from point B to the lower base AD parallel to the side trapezoids CD. Since BE and CD are parallel and drawn between parallel bases trapezoids BC and DA, then BCDE is a parallelogram and its opposite sides BE and CD are equal. BE=CD.
• Consider triangle ABE. Calculate side AE. AE=AD-ED. Reasons trapezoids BC and AD are known, and in parallelogram BCDE the opposite sides ED and BC are equal. ED=BC, so AE=AD-BC.
• Now find out the area of ​​triangle ABE using Heron's formula by calculating the semi-perimeter. S=root(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is the semi-perimeter of triangle ABE. p=1/2*(AB+BE+AE). To calculate the area, you know all the necessary data: AB, BE=CD, AE=AD-BC.
• Next, write down the area of ​​triangle ABE in a different way - it is equal to half the product of the height of triangle BH and the side AE ​​to which it is drawn. S=1/2*BH*AE.
• Express from this formula height triangle, which is also the height trapezoids. BH=2*S/AE. Calculate it.

If the trapezoid is isosceles, the solution can be done differently. Consider triangle ABH. It is rectangular because one of the corners, BHA, is right.

• Swipe from vertex C height CF.
• Study the HBCF figure. HBCF is a rectangle because two of its sides are heights and the other two are bases trapezoids, that is, the angles are right and the opposite sides are parallel. This means that BC=HF.
• Look at right triangles ABH and FCD. The angles at the heights BHA and CFD are right, and the angles at the sides BAH and CDF are equal, since the trapezoid ABCD is isosceles, which means the triangles are similar. Since the heights BH and CF are equal or the lateral sides of an isosceles trapezoids AB and CD are congruent, then similar triangles are congruent. This means that their sides AH and FD are also equal.
• Find AH. AH+FD=AD-HF. Since from a parallelogram HF=BC, and from triangles AH=FD, then AH=(AD-BC)*1/2.
• Next, from the right triangle ABH, using the Pythagorean theorem, calculate height B.H. Hypotenuse square AB equal to the sum squares of legs AH and BH. BH=root(AB*AB-AH*AH).

Trapeze is called a quadrilateral whose only two the sides are parallel to each other.

They are called the bases of the figure, the rest are called the sides. Parallelograms are considered special cases of the figure. There is also a curved trapezoid, which includes the graph of a function. Formulas for the area of ​​a trapezoid include almost all of its elements, and best solution is selected depending on the specified values.
The main roles in the trapezoid are assigned to the height and midline. Middle line- This is a line connecting the midpoints of the sides. Height The trapezoid is drawn at right angles from the top corner to the base.
The area of ​​a trapezoid through its height is equal to the product of half the sum of the lengths of the bases multiplied by the height:

If the average line is known according to the conditions, then this formula is significantly simplified, since it is equal to half the sum of the lengths of the bases:

If, according to the conditions, the lengths of all sides are given, then we can consider an example of calculating the area of ​​a trapezoid using these data:

Suppose we are given a trapezoid with bases a = 3 cm, b = 7 cm and sides c = 5 cm, d = 4 cm. Let’s find the area of ​​the figure:

Area of ​​an isosceles trapezoid

An isosceles trapezoid, or, as it is also called, an isosceles trapezoid, is considered a separate case.
A special case is finding the area of ​​an isosceles (equilateral) trapezoid. The formula is derived in various ways– through diagonals, through angles adjacent to the base and the radius of the inscribed circle.
If the length of the diagonals is specified according to the conditions and the angle between them is known, you can use the following formula:

Remember that the diagonals of an isosceles trapezoid are equal to each other!

That is, knowing one of their bases, side and angle, you can easily calculate the area.

Area of ​​a curved trapezoid

A special case is curved trapezoid. It is located on the coordinate axis and is limited by the graph of a continuous positive function.

Its base is located on the X axis and is limited to two points:
Integrals help calculate area curved trapezoid.
The formula is written like this:

Let's consider an example of calculating the area of ​​a curved trapezoid. The formula requires some knowledge to work with certain integrals. First, let's look at the value of the definite integral:

Here F(a) is the value antiderivative function f(x) at point a, F(b) is the value of the same function f(x) at point b.

Now let's solve the problem. The figure shows a curved trapezoid bounded by the function. Function
We need to find the area of ​​the selected figure, which is a curvilinear trapezoid bounded above by the graph, on the right by the straight line x =(-8), on the left by the straight line x =(-10) and the OX axis below.
We will calculate the area of ​​this figure using the formula:

The conditions of the problem give us a function. Using it we will find the values ​​of the antiderivative at each of our points:


Now
Answer: The area of ​​a given curved trapezoid is 4.

There is nothing complicated in calculating this value. The only thing that is important is extreme care in calculations.

There are many ways to find the area of ​​a trapezoid. Usually a math tutor knows several methods of calculating it, let’s look at them in more detail:
1) , where AD and BC are the bases, and BH is the height of the trapezoid. Proof: draw the diagonal BD and express the areas of triangles ABD and CDB through the half product of their bases and heights:

, where DP is the external height in

Let us add these equalities term by term and taking into account that the heights BH and DP are equal, we obtain:

Let's put it out of brackets

Q.E.D.

Corollary to the formula for the area of ​​a trapezoid:
Since the half-sum of the bases is equal to MN - the midline of the trapezoid, then

2) Application general formula area of ​​a quadrilateral.
The area of ​​a quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them
To prove it, it is enough to divide the trapezoid into 4 triangles, express the area of ​​each through “half the product of the diagonals and the sine of the angle between them” (taken as the angle, add the resulting expressions, take them out of the bracket and factor this bracket using the grouping method to obtain its equality to the expression. Hence

3) Diagonal shift method
This is my name. A math tutor will not come across such a heading in school textbooks. A description of the technique can only be found in additional textbooks as an example of solving a problem. I note that most of the interesting and useful facts mathematics tutors reveal planimetry to students in the process of performing practical work. This is extremely suboptimal, because the student needs to isolate them into separate theorems and call them “ big names" One of these is “diagonal shift”. About what we're talking about?Let us draw a line parallel to AC through vertex B until it intersects with the lower base at point E. In this case, the quadrilateral EBCA will be a parallelogram (by definition) and therefore BC=EA and EB=AC. The first equality is important to us now. We have:

Note that the triangle BED, whose area is equal to the area of ​​the trapezoid, has several more remarkable properties:
1) Its area is equal to the area of ​​the trapezoid
2) Its isosceles occurs simultaneously with the isosceles of the trapezoid itself
3) Its upper angle at vertex B is equal to the angle between the diagonals of the trapezoid (which is very often used in problems)
4) Its median BK is equal to the distance QS between the midpoints of the bases of the trapezoid. I recently came across the use of this property when preparing a student for Mechanics and Mathematics at Moscow State University using Tkachuk’s textbook, version 1973 (the problem is given at the bottom of the page).

Special techniques for a math tutor.

Sometimes I propose problems using a very tricky way of finding the area of ​​a trapezoid. I classify it as a special technique because in practice the tutor uses them extremely rarely. If you need preparation for the Unified State Exam in mathematics only in Part B, you don’t have to read about them. For others, I'll tell you further. It turns out that the area of ​​the trapezoid is twice more area a triangle with vertices at the ends of one side and the middle of the other, that is, the ABS triangle in the figure:
Proof: draw the heights SM and SN in triangles BCS and ADS and express the sum of the areas of these triangles:

Since point S is the midpoint of CD, then (prove it yourself). Find the sum of the areas of the triangles:

Since this sum turned out to be equal to half the area of ​​the trapezoid, then its second half. Etc.

I would include in the tutor’s collection of special techniques the form of calculating the area of ​​an isosceles trapezoid along its sides: where p is the semi-perimeter of the trapezoid. I won't give proof. Otherwise, your math tutor will be left without a job :). Come to class!

Problems on the area of ​​a trapezoid:

Math tutor's note: The list below is not a methodological accompaniment to the topic, it is only a small selection of interesting tasks based on the techniques discussed above.

1) The lower base of an isosceles trapezoid is 13, and the upper is 5. Find the area of ​​the trapezoid if its diagonal is perpendicular to the side.
2) Find the area of ​​a trapezoid if its bases are 2cm and 5cm, and its sides are 2cm and 3cm.
3) In an isosceles trapezoid, the larger base is 11, the side is 5, and the diagonal is Find the area of ​​the trapezoid.
4) The diagonal of an isosceles trapezoid is 5 and the midline is 4. Find the area.
5) In an isosceles trapezoid, the bases are 12 and 20, and the diagonals are mutually perpendicular. Calculate the area of ​​a trapezoid
6) The diagonal of an isosceles trapezoid makes an angle with its lower base. Find the area of ​​the trapezoid if its height is 6 cm.
7) The area of ​​the trapezoid is 20, and one of its sides is 4 cm. Find the distance to it from the middle of the opposite side.
8) The diagonal of an isosceles trapezoid divides it into triangles with areas of 6 and 14. Find the height if the lateral side is 4.
9) In a trapezoid, the diagonals are equal to 3 and 5, and the segment connecting the midpoints of the bases is equal to 2. Find the area of ​​the trapezoid (Mekhmat MSU, 1970).

I chose not the most difficult problems (don’t be afraid of mechanical engineering!) with the expectation that I would be able to solve them independently. Decide for your health! If you need preparation for the Unified State Exam in mathematics, then without participation in this process, formulas for the area of ​​a trapezoid may arise serious problems even with problem B6 and even more so with C4. Do not start the topic and in case of any difficulties, ask for help. A math tutor is always happy to help you.

Kolpakov A.N.
Mathematics tutor in Moscow, preparation for the Unified State Exam in Strogino.

A trapezoid is a relief quadrilateral in which two opposite sides are parallel and the other two are non-parallel. If all opposite sides of a quadrilateral are parallel in pairs, then it is a parallelogram.

You will need

• – all sides of the trapezoid (AB, BC, CD, DA).

Instructions

1. Non-parallel sides trapezoids are called lateral sides, and parallel sides are called bases. The line between the bases, perpendicular to them - height trapezoids. If lateral sides trapezoids are equal, then it is called isosceles. First, let's look at the solution for trapezoids, which is not isosceles.

2. Draw line segment BE from point B to the lower base AD parallel to the side trapezoids CD. Because BE and CD are parallel and drawn between parallel bases trapezoids BC and DA, then BCDE is a parallelogram, and its opposite sides BE and CD are equal. BE=CD.

3. Look at the triangle ABE. Calculate side AE. AE=AD-ED. Reasons trapezoids BC and AD are known, and in a parallelogram BCDE are opposite sides ED and BC are equal. ED=BC, so AE=AD-BC.

4. Now find out the area of ​​triangle ABE using Heron's formula by calculating the semi-perimeter. S=root(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is the semi-perimeter of triangle ABE. p=1/2*(AB+BE+AE). To calculate the area, you know all the necessary data: AB, BE=CD, AE=AD-BC.

6. Express from this formula the height of the triangle, which is also the height trapezoids. BH=2*S/AE. Calculate it.

7. If the trapezoid is isosceles, the solution can be executed differently. Look at the triangle ABH. It is rectangular because one of the corners, BHA, is right.

8. Draw height CF from vertex C.

9. Study the HBCF figure. HBCF rectangle, because there are two of it sides are heights, and the other two are bases trapezoids, that is, the angles are right, and the opposite sides parallel. This means that BC=HF.

10. Look at the right triangles ABH and FCD. The angles at heights BHA and CFD are right, and the angles at lateral sides x BAH and CDF are equal because the trapezoid ABCD is isosceles, which means the triangles are similar. Because the heights BH and CF are equal or lateral sides isosceles trapezoids AB and CD are congruent, then similar triangles are congruent. So they sides AH and FD are also equal.

11. Discover AH. AH+FD=AD-HF. Because from a parallelogram HF=BC, and from triangles AH=FD, then AH=(AD-BC)*1/2.

Trapezoid – geometric figure, which is a quadrilateral in which two sides, called bases, are parallel, and the other two are not parallel. They are called sides trapezoids. The segment drawn through the midpoints of the lateral sides is called the midline trapezoids. A trapezoid can have different side lengths or identical ones, in which case it is called isosceles. If one of the sides is perpendicular to the base, then the trapezoid will be rectangular. But it is much more practical to know how to detect square trapezoids .

You will need

• Ruler with millimeter graduations

Instructions

1. Measure all sides trapezoids: AB, BC, CD and DA. Record your measurements.

2. On segment AB, mark the middle - point K. On segment DA, mark point L, which is also in the middle of segment AD. Combine points K and L, the resulting segment KL will be the middle line trapezoids ABCD. Measure the segment KL.

3. From the top trapezoids– toss C, lower the perpendicular to its base AD on the segment CE. It will be the height trapezoids ABCD. Measure the segment CE.

4. Let us call the segment KL the letter m, and the segment CE the letter h, then square S trapezoids ABCD is calculated using the formula: S=m*h, where m is the middle line trapezoids ABCD, h – height trapezoids ABCD.

5. There is another formula that allows you to calculate square trapezoids ABCD. Bottom base trapezoids– Let’s call AD the letter b, and the upper base BC the letter a. The area is determined by the formula S=1/2*(a+b)*h, where a and b are the bases trapezoids, h – height trapezoids .

Video on the topic

Tip 3: How to find the height of a trapezoid if the area is known

A trapezoid is a quadrilateral in which two of its four sides are parallel to each other. Parallel sides are the bases of this trapezoids, the other two are the lateral sides of this trapezoids. Discover height trapezoids, if you know its area, it will be very easy.

Instructions

1. We need to figure out how to calculate the area of ​​the initial trapezoids. There are several formulas for this, depending on the initial data: S = ((a+b)*h)/2, where a and b are the lengths of the bases trapezoids, and h is its height (Height trapezoids– perpendicular, lowered from one base trapezoids to another);S = m*h, where m is the middle line trapezoids(The middle line is a segment parallel to the bases trapezoids and connecting the midpoints of its sides).

2. Now, knowing the formulas for calculating area trapezoids, it is allowed to derive new ones from them to find the height trapezoids:h = (2*S)/(a+b);h = S/m.

3. In order to make it clearer how to solve similar problems, you can look at examples: Example 1: Given a trapezoid whose area is 68 cm?, the middle line of which is 8 cm, you need to find height given trapezoids. In order to solve this problem, you need to use the previously derived formula: h = 68/8 = 8.5 cm Answer: the height of this trapezoids is 8.5 cmExample 2: Let y trapezoids area is 120 cm?, the length of the bases is given trapezoids are equal to 8 cm and 12 cm respectively, it is required to detect height this trapezoids. To do this, you need to apply one of the derived formulas: h = (2*120)/(8+12) = 240/20 = 12 cmAnswer: height of the given trapezoids equal to 12 cm

Video on the topic

Pay attention!
Any trapezoid has a number of properties: - the middle line of a trapezoid is equal to half the sum of its bases; - the segment that connects the diagonals of the trapezoid is equal to half the difference of its bases; - if a straight line is drawn through the midpoints of the bases, then it will intersect the point of intersection of the diagonals of the trapezoid; - You can inscribe a circle into a trapezoid if the sum of the bases of a given trapezoid is equal to the sum of its sides. Use these properties when solving problems.

Tip 4: How to find the height of a triangle given the coordinates of the points

The height in a triangle is the straight line segment connecting the vertex of the figure to the opposite side. This segment must certainly be perpendicular to the side; therefore, from any vertex it is allowed to draw only one height. Because there are three vertices in this figure, there are the same number of heights. If a triangle is given by the coordinates of its vertices, the length of each of the heights can be calculated, say, using the formula for finding the area and calculating the lengths of the sides.

Instructions

1. Proceed in your calculations from the fact that the area triangle is equal to half the product of the length of each of its sides by the length of the height lowered onto this side. From this definition it follows that to find the height you need to know the area of ​​the figure and the length of the side.

2. Start by calculating the lengths of the sides triangle. Designate the coordinates of the vertices of the figure as follows: A(X?,Y?,Z?), B(X?,Y?,Z?) and C(X?,Y?,Z?). Then you can calculate the length of side AB using the formula AB = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?). For the other 2 sides, these formulas will look like this: BC = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?) and AC = ?((X ?-X?)? + (Y?-Y?)? + (Z?-Z?)?). Let's say for triangle with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) ​​the length of side AB will be?((3-16)? + (5-14)? + (7 -19)?) = ?(-13? + (-9?) + (-12?)) = ?(169 + 81 + 144) = ?394 ? 19.85. The lengths of the sides BC and AC, calculated by the same method, will be equal?(15? + 12? + 6?) = ?405? 20.12 and?(2? + 3? + (-6?)) =?49 = 7.

3. Knowing the lengths of 3 sides obtained in the previous step is enough to calculate the area triangle(S) according to Heron’s formula: S = ? * ?((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). Let's say, after substituting into this formula the values ​​​​obtained from the coordinates triangle-example from the previous step, this formula will give the following value: S = ?*?((19.85+20.12+7) * (20.12+7-19.85) * (19.85+7-20 .12) * (19.85+20.12-7)) = ?*?(46.97 * 7.27 * 6.73 * 32.97) ? ?*?75768.55 ? ?*275.26 = 68.815.

4. Based on area triangle, calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the heights for each of the sides. Because the area is equal to half the product of the height and the length of the side to which it is drawn, to find the height, divide the doubled area by the length of the required side: H = 2*S/a. For the example used above, the height lowered to side AB will be 2*68.815/16.09? 8.55, the height to the BC side will have a length of 2*68.815/20.12? 6.84, and for the AC side this value will be equal to 2*68.815/7? 19.66.

The practice of last year's Unified State Exam and State Examination shows that geometry problems cause difficulties for many schoolchildren. You can easily cope with them if you memorize all the necessary formulas and practice solving problems.

In this article you will see formulas for finding the area of ​​a trapezoid, as well as examples of problems with solutions. You may come across the same ones in KIMs during certification exams or at Olympiads. Therefore, treat them carefully.

What you need to know about the trapezoid?

To begin with, let us remember that trapezoid is called a quadrilateral in which two opposite sides, also called bases, are parallel, and the other two are not.

In a trapezoid, the height (perpendicular to the base) can also be lowered. The middle line is drawn - this is a straight line that is parallel to the bases and equal to half of their sum. As well as diagonals that can intersect, forming acute and obtuse angles. Or, in some cases, at a right angle. In addition, if the trapezoid is isosceles, a circle can be inscribed in it. And describe a circle around it.

Trapezoid area formulas

First, let's look at the standard formulas for finding the area of ​​a trapezoid. We will consider ways to calculate the area of ​​isosceles and curvilinear trapezoids below.

So, imagine that you have a trapezoid with bases a and b, in which height h is lowered to the larger base. Calculating the area of ​​a figure in this case is as easy as shelling pears. You just need to divide the sum of the lengths of the bases by two and multiply the result by the height: S = 1/2(a + b)*h.

Let's take another case: suppose in a trapezoid, in addition to the height, there is a middle line m. We know the formula for finding the length of the middle line: m = 1/2(a + b). Therefore, we can rightfully simplify the formula for the area of ​​a trapezoid to the following type: S = m* h. In other words, to find the area of ​​a trapezoid, you need to multiply the center line by the height.

Let's consider another option: the trapezoid contains diagonals d 1 and d 2, which do not intersect at right angles α. To calculate the area of ​​such a trapezoid, you need to divide the product of the diagonals by two and multiply the result by the sin of the angle between them: S= 1/2d 1 d 2 *sinα.

Now consider the formula for finding the area of ​​a trapezoid if nothing is known about it except the lengths of all its sides: a, b, c and d. This is a cumbersome and complex formula, but it will be useful for you to remember it just in case: S = 1/2(a + b) * √c 2 – ((1/2(b – a)) * ((b – a) 2 + c 2 – d 2)) 2.

By the way, the above examples are also true for the case when you need the formula for the area of ​​a rectangular trapezoid. This is a trapezoid, the side of which adjoins the bases at a right angle.

Isosceles trapezoid

A trapezoid whose sides are equal is called isosceles. We will consider several options for the formula for the area of ​​an isosceles trapezoid.

First option: for the case when a circle with radius r is inscribed inside an isosceles trapezoid, and the side and larger base form acute angleα. A circle can be inscribed in a trapezoid provided that the sum of the lengths of its bases is equal to the sum of the lengths of the sides.

The area of ​​an isosceles trapezoid is calculated as follows: multiply the square of the radius of the inscribed circle by four and divide it all by sinα: S = 4r 2 /sinα. Another area formula is a special case for the option when the angle between the large base and the side is 30 0: S = 8r2.

Second option: this time we take an isosceles trapezoid, in which in addition the diagonals d 1 and d 2 are drawn, as well as the height h. If the diagonals of a trapezoid are mutually perpendicular, the height is half the sum of the bases: h = 1/2(a + b). Knowing this, it is easy to transform the formula for the area of ​​a trapezoid already familiar to you into this form: S = h 2.

Formula for the area of ​​a curved trapezoid

Let's start by figuring out what a curved trapezoid is. Imagine a coordinate axis and a graph of a continuous and non-negative function f that does not change sign within a given segment on the x-axis. A curvilinear trapezoid is formed by the graph of the function y = f(x) - at the top, the x axis is at the bottom (segment), and on the sides - straight lines drawn between points a and b and the graph of the function.

It is impossible to calculate the area of ​​such a non-standard figure using the above methods. Here you need to apply mathematical analysis and use the integral. Namely: the Newton-Leibniz formula - S = ∫ b a f(x)dx = F(x)│ b a = F(b) – F(a). In this formula, F is the antiderivative of our function on the selected segment. And the area of ​​a curvilinear trapezoid corresponds to the increment of the antiderivative on a given segment.

Sample problems

To make all these formulas easier to understand in your head, here are some examples of problems for finding the area of ​​a trapezoid. It will be best if you first try to solve the problems yourself, and only then compare the answer you receive with the ready-made solution.

Task #1: Given a trapezoid. Its larger base is 11 cm, the smaller one is 4 cm. The trapezoid has diagonals, one 12 cm long, the second 9 cm.

Solution: Construct a trapezoid AMRS. Draw a straight line РХ through vertex P so that it is parallel to the diagonal MC and intersects the straight line AC at point X. You will get a triangle APХ.

We will consider two figures obtained as a result of these manipulations: triangle APX and parallelogram CMRX.

Thanks to the parallelogram, we learn that PX = MC = 12 cm and CX = MR = 4 cm. From where we can calculate the side AX of the triangle ARX: AX = AC + CX = 11 + 4 = 15 cm.

We can also prove that the triangle APX is right-angled (to do this, apply the Pythagorean theorem - AX 2 = AP 2 + PX 2). And calculate its area: S APX = 1/2(AP * PX) = 1/2(9 * 12) = 54 cm 2.

Next you will need to prove that triangles AMP and PCX are equal in area. The basis will be the equality of the parties MR and CX (already proven above). And also the heights that you lower on these sides - they are equal to the height of the AMRS trapezoid.

All this will allow you to say that S AMPC = S APX = 54 cm 2.

Task #2: The trapezoid KRMS is given. On its lateral sides there are points O and E, while OE and KS are parallel. It is also known that the areas of the trapezoids ORME and OKSE are in the ratio 1:5. RM = a and KS = b. You need to find OE.

Solution: Draw a line parallel to RK through point M, and designate the point of its intersection with OE as T. A is the point of intersection of a line drawn through point E parallel to RK with the base KS.

Let's introduce one more notation - OE = x. And also the height h 1 for the triangle TME and the height h 2 for the triangle AEC (you can independently prove the similarity of these triangles).

We will assume that b > a. The areas of the trapezoids ORME and OKSE are in the ratio 1:5, which gives us the right to create the following equation: (x + a) * h 1 = 1/5(b + x) * h 2. Let's transform and get: h 1 / h 2 = 1/5 * ((b + x)/(x + a)).

Since the triangles TME and AEC are similar, we have h 1 / h 2 = (x – a)/(b – x). Let's combine both entries and get: (x – a)/(b – x) = 1/5 * ((b + x)/(x + a)) ↔ 5(x – a)(x + a) = (b + x)(b – x) ↔ 5(x 2 – a 2) = (b 2 – x 2) ↔ 6x 2 = b 2 + 5a 2 ↔ x = √(5a 2 + b 2)/6.

Thus, OE = x = √(5a 2 + b 2)/6.

Conclusion

Geometry is not the easiest of sciences, but you can certainly cope with the exam questions. It is enough to show a little perseverance in preparation. And, of course, remember all the necessary formulas.

We tried to collect all the formulas for calculating the area of ​​a trapezoid in one place so that you can use them when you prepare for exams and revise the material.

Be sure to tell your classmates and friends about this article. social networks. Let there be more good grades for the Unified State Examination and State Examinations!

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