In an isosceles trapezoid, the sums of opposite sides are equal. Circumscribed circle and trapezium
A trapezoid is a convex quadrilateral in which one pair of opposite sides are parallel to each other, and the other is not.
Based on the definition of a trapezoid and the features of a parallelogram, the parallel sides of a trapezoid cannot be equal to each other. Otherwise, the other pair of sides would also become parallel and equal to each other. In this case, we would be dealing with a parallelogram.
Parallel opposite sides of a trapezoid are called grounds. That is, a trapezoid has two bases. Non-parallel opposite sides of a trapezoid are called sides.
Depending on which sides, what angles they form with the bases, they distinguish different kinds trapezium. Most often, trapezoids are divided into non-isosceles (diverse), isosceles (isolateral) and rectangular.
At lateral trapezoids sides are not equal. At the same time, with a large base, they can both form only acute angles, or one angle will be obtuse and the other acute. In the first case, the trapezoid is called acute-angled, in the second - obtuse.
At isosceles trapezoids sides are equal to each other. At the same time, with a large base, they can form only sharp corners, i.e. All isosceles trapezoids are acute-angled. Therefore, they are not divided into acute-angled and obtuse-angled.
At rectangular trapezoid one side perpendicular to the bases. The second side cannot be perpendicular to them, because in this case we would be dealing with a rectangle. In rectangular trapezoids, the non-perpendicular side always forms an acute angle with a large base. The perpendicular side is perpendicular to both bases, since the bases are parallel.
Instruction
According to the property of an isosceles trapezoid, the segment n is equal to the half-difference of the bases x and y. Therefore, the smaller base of the trapezoid y can be represented as the difference between the larger base and the segment n multiplied by two: y \u003d x - 2 * n.
Find the unknown smaller segment n. To do this, calculate one of the sides of the resulting right triangle. The triangle is formed by a height - h (leg), a side - a (hypotenuse) and a segment - n (leg). According to the Pythagorean theorem, the unknown leg is n² = a² - h². Substitute the numerical values and calculate the square of the leg n. Take the square root of the resulting value - this will be the length of the segment n.
Substitute the resulting value into the first equation to calculate y. The area of the trapezoid is calculated by the formula S = ((x + y)*h)/2. Express the unknown variable: y = 2*S/h - x.
Sources:
- height of an isosceles trapezoid
To define a quadrangle such as a trapezoid, at least three of its sides must be defined. Therefore, for example, we can consider a problem in which the lengths of the diagonals are given trapeze, as well as one of the lateral side vectors.
Instruction
The figure from the condition of the problem is presented on 1. In this case, it should be assumed that the one under consideration is ABCD, in which the lengths of the diagonals AC and BD are given, as well as the side of AB, represented by the vector a(ax,ay). The accepted initial data allow us to find both grounds trapeze(both top and bottom). IN specific example the lower base AD will be found first.
Consider triangle ABD. The length of its side AB is equal to the modulus of the vector a. Let|a|=sqrt((ax)^2+(ay)^2)=a, then cosФ =ax/sqrt(((ax)^2+(ay)^2) as the direction cosine of a. Let the given diagonal BD has length p, and the desired AD length X. Then, according to the cosine theorem, P^2=a^2+ x^2-2axcosph. Or x^2-2axcosf+(a^2-p^2)=0.
To find the top grounds BC (its length in the search is also indicated by x), the module |a|=a is used, as well as the second diagonal BD=q and the cosine of the angle ABC, which, obviously, is equal to (pf).
Next, the triangle ABC is considered, to which, as before, the cosine theorem, and the following arises. Given that cos(pf)=-cosph, based on the solution for AD, the following formula can be obtained by replacing p with q: BC=- a*ax|sqrt(((ax)^2+(ay)^2) +sqrt((((a)^2)(ax^2))/(ax^2+ay^2))-a^2+q^2).
This is a square and, accordingly, has two roots. Thus, in this case, it remains to choose only those roots that have positive value, since the length cannot be negative.
ExampleLet in trapeze ABCD side AB is given by vector a(1, sqrt3), p=4, q=6. To find grounds trapeze.Solution. Using the algorithms obtained above, we can write: |a|=a=2, cosf=1/2. AD=1/2+sqrt(4/4 -4+16)=1/2 +sqrt(13)=(sqrt(13)+1)/2.BC=-1/2+sqrt(-3+36 )=(sqrt(33)-1)/2.
Related videos
A trapezoid is a quadrilateral in which two sides are parallel and the other two are not. The height of a trapezoid is a segment drawn perpendicularly between two parallel lines. Depending on the source data, it can be calculated in different ways.
You will need
- Knowledge of the sides, bases, midline of a trapezoid, and optionally its area and/or perimeter.
Instruction
Let's say there is a trapezoid with the same data as in Figure 1. Let's draw 2 heights, we get, which has 2 smaller sides with the legs of right triangles. Let's denote the smaller roll as x. It is found by dividing the difference in length between the larger and smaller bases. Then, according to the Pythagorean theorem, the square of the height is equal to the sum of the squares of the hypotenuse d and the leg x. Extract from this sum and get the height h. (Fig. 2)
Related videos
Sources:
- how to calculate the height of a trapezoid
A mathematical figure with four corners is called a trapezoid if a pair of its opposite sides are parallel and the other pair is not. Parallel sides are called grounds trapeze, the other two are lateral. In a rectangular trapeze one of the corners at the lateral side is straight.
Instruction
Task 1. Find the bases of BC and AD trapeze, if the length AC = f is known; side length CD = c and its angle ADC = α. Solution: Consider a rectangular CED. The hypotenuse c and the angle between the hypotenuse and the leg EDC are known. Find the lengths of CE and ED: using the angle formula CE = CD*sin(ADC); ED=CD*cos(ADC). So: CE = c*sinα; ED=c*cosα.
Consider right triangle ACE. You know the hypotenuse AC and CE, find the side AE according to the rule: the sum of the squares of the legs is equal to the square of the hypotenuse. So: AE(2) = AC(2) - CE(2) = f(2) - c*sinα. Calculate Square root from the right side of the equality. You found the top rectangular trapeze.
The length of the base AD is the sum of the lengths of the two segments AE and ED. AE = square root(f(2) - c*sinα); ED = c*cosα). So: AD = square root(f(2) - c*sinα) + c*cosα. You have found the lower base of the rectangular trapeze.
Task 2. Find the bases BC and AD of a rectangular trapeze, if the length of the diagonal is known BD = f; side length CD = c and its angle ADC = α. Solution: Consider a right triangle CED. Find the lengths of the sides CE and ED: CE = CD*sin(ADC) = c*sinα; ED = CD*cos(ADC) = c*cosα.
Consider the rectangle ABCE. By property AB = CE = c*sinα. Consider a right triangle ABD. According to the property of a right triangle, the square of the hypotenuse is the sum of the squares of the legs. Therefore AD(2) = BD(2) - AB(2) = f(2) - c*sinα. You have found the lower base of the rectangular trapeze AD = square root(f(2) - c*sinα).
By the rectangle rule BC = AE = AD - ED = square root(f(2) - c*sinα) - c*cosα. You have found the upper base of the rectangular trapeze.
The smaller base of a trapezoid is one of its parallel sides, which has a minimum length. This value can be calculated in several ways, using certain data.
You will need
- - calculator.
Instruction
If two lengths are known - the base and the midline - use the property of the trapezoid to calculate the smallest base. According to him, the middle line of a trapezoid is identical to half the sum of the bases. In this case, the smallest base will be equal to the difference between the doubled length of the midline and the length of the large base of this figure.
If such parameters of the trapezoid as, height, length of the large base are known, then the calculation of the smallest base of this lead on the basis of the trapezoid. In this case, get the final result by subtracting from the difference of the quotient twice the area and the height of such a parameter as the length of the large base of the trapezoid.
Calculate the length of the side in the other
With such a form as a trapezoid, we meet in life quite often. For example, any bridge that is made of concrete blocks is a prime example. A more visual option can be considered the steering of each vehicle And so on. The properties of the figure were already known in Ancient Greece
, which was described in more detail by Aristotle in his scientific work"Start". And the knowledge that was developed thousands of years ago is still relevant today. Therefore, we will get acquainted with them in more detail.
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Basic concepts
Figure 1. The classic shape of a trapezoid.
A trapezoid is essentially a quadrilateral, consisting of two segments that are parallel and two others that are not parallel. Speaking about this figure, it is always necessary to remember such concepts as: bases, height and middle line. Two segments of a quadrilateral which are called bases to each other (segments AD and BC). The height is called the segment perpendicular to each of the bases (EH), i.e. intersect at an angle of 90° (as shown in Fig. 1).
If we add up all the degree measures of the internal, then the sum of the angles of the trapezoid will be equal to 2π (360 °), like any quadrilateral. A segment whose ends are the midpoints of the sidewalls (IF) called the middle line. The length of this segment is the sum of the bases BC and AD divided by 2.
There are three types geometric figure: straight, regular and isosceles. If at least one angle at the vertices of the base is right (for example, if ABD = 90 °), then such a quadrilateral is called a right trapezoid. If the side segments are equal (AB and CD), then it is called isosceles (respectively, the angles at the bases are equal).
How to find the area
For that, to find the area of a quadrilateral ABCD use the following formula:
Figure 2. Solving the problem of finding the area
For a more illustrative example, let's solve an easy problem. For example, let the upper and lower bases be equal to 16 and 44 cm, respectively, and the sides are 17 and 25 cm. Let's build a perpendicular segment from the vertex D so that DE II BC (as shown in Figure 2). Hence we get that
Let DF - will be. From ΔADE (which will be equilateral), we get the following:
That is, to express plain language, we first found the height ΔADE, which is also the height of the trapezoid. From here we calculate the area of the quadrilateral ABCD, with the already known value of the height DF, using the already known formula.
Hence, the desired area ABCD is 450 cm³. That is, it can be said with certainty that To calculate the area of a trapezoid, you need only the sum of the bases and the length of the height.
Important! When solving the problem, it is not necessary to find the value of the lengths separately; it is quite possible if other parameters of the figure are applied, which, with appropriate proof, will be equal to the sum of the bases.
Types of trapezium
Depending on which sides the figure has, what angles are formed at the bases, there are three types of quadrilateral: rectangular, sided and equilateral.
Versatile
There are two forms: acute and obtuse. ABCD is acute only if the base angles (AD) are acute and the side lengths are different. If the value of one angle is the number Pi / 2 more (the degree measure is more than 90 °), then we get an obtuse angle.
If the sides are equal in length
Figure 3. View of an isosceles trapezoid
If non-parallel sides are equal in length, then ABCD is called isosceles (correct). Moreover, for such a quadrilateral, the degree measure of the angles at the base is the same, their angle will always be less than the right one. It is for this reason that the isosceles is never divided into acute and obtuse. A quadrilateral of this shape has its own specific differences, which include:
- The segments connecting opposite vertices are equal.
- Acute angles with a larger base are 45 ° (an illustrative example in Figure 3).
- If you add the degrees of opposite angles, then in total they will give 180 °.
- Around any regular trapezoid can be built.
- If you add the degree measure of opposite angles, then it is equal to π.
Moreover, due to their geometric arrangement of points, there are basic properties of an isosceles trapezoid:
Angle value at base 90°
The perpendicularity of the lateral side of the base is a capacious characteristic of the concept of "rectangular trapezium". There cannot be two sides with corners at the base, because otherwise it will be already a rectangle. In quadrilaterals of this type, the second side will always form an acute angle with a large base, and with a smaller one - obtuse. In this case, the perpendicular side will also be the height.
Segment between the middle of the sidewalls
If we connect the midpoints of the sides, and the resulting segment will be parallel to the bases, and equal in length to half their sum, then the formed straight line will be the middle line. The value of this distance is calculated by the formula:
For a more illustrative example, consider a problem using the middle line.
A task. The median line of the trapezoid is 7 cm, it is known that one of the sides is 4 cm larger than the other (Fig. 4). Find the lengths of the bases.
Figure 4. Solving the problem of finding base lengths
Solution. Let the smaller base of DC be equal to x cm, then the larger base will be equal to (x + 4) cm, respectively. From here, using the formula for the middle line of the trapezoid, we get:
It turns out that the smaller base of DC is 5 cm, and the larger one is 9 cm.
Important! The concept of the median line is the key to solving many problems in geometry. Based on its definition, many proofs for other figures are built. Using the concept in practice, perhaps more rational solution and search for the required value.
Determination of height, and how to find it
As noted earlier, the height is a segment that intersects the bases at an angle of 2Pi / 4 and is the shortest distance between them. Before finding the height of the trapezoid, it is necessary to determine what input values are given. For better understanding consider the problem. Find the height of the trapezoid, provided that the bases are 8 and 28 cm, the sides are 12 and 16 cm, respectively.
Figure 5. Solving the problem of finding the height of a trapezoid
Let's draw segments DF and CH at right angles to the base AD. According to the definition, each of them will be the height of a given trapezoid (Fig. 5). In this case, knowing the length of each sidewall, using the Pythagorean theorem, we find what the height in triangles AFD and BHC is.
The sum of the segments AF and HB is equal to the difference of the bases, i.e.:
Let the length of AF be equal to x cm, then the length of the segment HB = (20 - x) cm. As it was established, DF=CH , hence .
Then we get the following equation:
It turns out that the segment AF in the triangle AFD is 7.2 cm, from here we calculate the height of the trapezoid DF using the same Pythagorean theorem:
Those. the height of the ADCB trapezoid will be 9.6 cm. As you can see, the height calculation is a more mechanical process, and is based on the calculations of the sides and angles of triangles. But, in a number of problems in geometry, only degrees of angles can be known, in which case the calculations will be made through the ratio of the sides of the inner triangles.
Important! In essence, a trapezoid is often thought of as two triangles, or as a combination of a rectangle and a triangle. To solve 90% of all problems found in school textbooks, the properties and characteristics of these figures. Most of the formulas for this GMT are derived relying on the "mechanisms" for these two types of figures.
How to quickly calculate the length of the base
Before you find the base of the trapezoid, you need to determine what parameters are already given, and how to use them rationally. A practical approach is to extract the length of the unknown base from the midline formula. For a clearer perception of the picture, we will show how this can be done using an example of a task. Let it be known that the middle line of the trapezoid is 7 cm, and one of the bases is 10 cm. Find the length of the second base.
Solution: Knowing that the middle line is equal to half the sum of the bases, it can be argued that their sum is 14 cm.
(14cm=7cm×2). From the condition of the problem, we know that one of is equal to 10 cm, hence the smaller side of the trapezoid will be equal to 4 cm (4 cm = 14 - 10).
Moreover, for a more comfortable solution of problems of this kind, we recommend that you learn well such formulas from the trapezoid area as:
- middle line;
- area;
- height;
- diagonals.
Knowing the essence (precisely the essence) of these calculations, you can easily find out the desired value.
Video: trapezium and its properties
Video: trapezoid features
Output
From the considered examples of problems, we can draw a simple conclusion that the trapezoid, in terms of calculating problems, is one of the simplest figures in geometry. To successfully solve problems, first of all, it is not necessary to decide what information is known about the object being described, in what formulas they can be applied, and decide what needs to be found. By executing this simple algorithm, no task using this geometric figure will be effortless.
In this article, we will try to reflect the properties of the trapezoid as fully as possible. In particular, we will talk about common features and properties of a trapezoid, as well as about the properties of an inscribed trapezoid and about a circle inscribed in a trapezoid. We will also touch on the properties of an isosceles and rectangular trapezoid.
An example of solving a problem using the considered properties will help you sort things out in your head and better remember the material.
Trapeze and all-all-all
To begin with, let's briefly recall what a trapezoid is and what other concepts are associated with it.
So, a trapezoid is a quadrilateral figure, two of the sides of which are parallel to each other (these are the bases). And two are not parallel - these are the sides.
In a trapezoid, the height can be omitted - perpendicular to the bases. The middle line and diagonals are drawn. And also from any angle of the trapezoid it is possible to draw a bisector.
Pro various properties associated with all these elements and their combinations, we will talk now.
Properties of the diagonals of a trapezoid
To make it clearer, while reading, sketch out the ACME trapezoid on a piece of paper and draw diagonals in it.
- If you find the midpoints of each of the diagonals (let's call these points X and T) and connect them, you get a segment. One of the properties of the diagonals of a trapezoid is that the segment XT lies on the midline. And its length can be obtained by dividing the difference of the bases by two: XT \u003d (a - b) / 2.
- Before us is the same ACME trapezoid. The diagonals intersect at point O. Let's consider the triangles AOE and IOC formed by the segments of the diagonals together with the bases of the trapezoid. These triangles are similar. The similarity coefficient of k triangles is expressed in terms of the ratio of the bases of the trapezoid: k = AE/KM.
The ratio of the areas of triangles AOE and IOC is described by the coefficient k 2 . - All the same trapezium, the same diagonals intersecting at point O. Only this time we will consider triangles that the diagonal segments formed together with the sides of the trapezoid. The areas of triangles AKO and EMO are equal - their areas are the same.
- Another property of a trapezoid includes the construction of diagonals. So, if we continue the sides of AK and ME in the direction of the smaller base, then sooner or later they will intersect to some point. Next, draw a straight line through the midpoints of the bases of the trapezoid. It intersects the bases at points X and T.
If we now extend the line XT, then it will join together the point of intersection of the diagonals of the trapezoid O, the point at which the extensions of the sides and the midpoints of the bases of X and T intersect. - Through the point of intersection of the diagonals, we draw a segment that will connect the bases of the trapezoid (T lies on the smaller base of KM, X - on the larger AE). The intersection point of the diagonals divides this segment in the following ratio: TO/OH = KM/AE.
- And now through the point of intersection of the diagonals we draw a segment parallel to the bases of the trapezoid (a and b). The intersection point will divide it into two equal parts. You can find the length of a segment using the formula 2ab/(a + b).
Properties of the midline of a trapezoid
Draw the middle line in the trapezium parallel to its bases.
- The length of the midline of a trapezoid can be calculated by adding the lengths of the bases and dividing them in half: m = (a + b)/2.
- If you draw any segment (height, for example) through both bases of the trapezoid, the middle line will divide it into two equal parts.
Property of the bisector of a trapezoid
Pick any angle of the trapezoid and draw a bisector. Take, for example, the angle KAE of our trapezoid ACME. Having completed the construction on your own, you can easily see that the bisector cuts off from the base (or its continuation on a straight line outside the figure itself) a segment of the same length as the side.
Trapezoid angle properties
- Whichever of the two pairs of angles adjacent to the side you choose, the sum of the angles in a pair is always 180 0: α + β = 180 0 and γ + δ = 180 0 .
- Connect the midpoints of the bases of the trapezoid with a segment TX. Now let's look at the angles at the bases of the trapezoid. If the sum of the angles for any of them is 90 0, the length of the TX segment is easy to calculate based on the difference in the lengths of the bases, divided in half: TX \u003d (AE - KM) / 2.
- If parallel lines are drawn through the sides of the angle of a trapezoid, they will divide the sides of the angle into proportional segments.
Properties of an isosceles (isosceles) trapezoid
- In an isosceles trapezoid, the angles at any of the bases are equal.
- Now build a trapezoid again to make it easier to imagine what it is about. Look carefully at the base of AE - the vertex of the opposite base of M is projected to a certain point on the line that contains AE. The distance from vertex A to the projection point of vertex M and the midline of an isosceles trapezoid are equal.
- A few words about the property of the diagonals of an isosceles trapezoid - their lengths are equal. And also the angles of inclination of these diagonals to the base of the trapezoid are the same.
- Only near an isosceles trapezoid can a circle be described, since the sum of the opposite angles of a quadrilateral 180 0 is a prerequisite for this.
- The property of an isosceles trapezoid follows from the previous paragraph - if a circle can be described near a trapezoid, it is isosceles.
- From the features of an isosceles trapezoid, the property of the height of a trapezoid follows: if its diagonals intersect at a right angle, then the length of the height is equal to half the sum of the bases: h = (a + b)/2.
- Draw the line TX again through the midpoints of the bases of the trapezoid - in an isosceles trapezoid it is perpendicular to the bases. And at the same time, TX is the axis of symmetry of an isosceles trapezoid.
- This time lower to the larger base (let's call it a) the height from the opposite vertex of the trapezoid. You will get two cuts. The length of one can be found if the lengths of the bases are added and divided in half: (a+b)/2. We get the second one when we subtract the smaller one from the larger base and divide the resulting difference by two: (a – b)/2.
Properties of a trapezoid inscribed in a circle
Since we are already talking about a trapezoid inscribed in a circle, let's dwell on this issue in more detail. In particular, where is the center of the circle in relation to the trapezoid. Here, too, it is recommended not to be too lazy to pick up a pencil and draw what will be discussed below. So you will understand faster, and remember better.
- The location of the center of the circle is determined by the angle of inclination of the diagonal of the trapezoid to its side. For example, a diagonal may emerge from the top of a trapezoid at right angles to the side. In this case, the larger base intersects the center of the circumscribed circle exactly in the middle (R = ½AE).
- The diagonal and the side can also meet at an acute angle - then the center of the circle is inside the trapezoid.
- The center of the circumscribed circle may be outside the trapezium, beyond its large base, if there is an obtuse angle between the diagonal of the trapezoid and the lateral side.
- The angle formed by the diagonal and the large base of the trapezoid ACME (inscribed angle) is half that central corner, which corresponds to it: MAE = ½MY.
- Briefly about two ways to find the radius of the circumscribed circle. Method one: look carefully at your drawing - what do you see? You will easily notice that the diagonal splits the trapezoid into two triangles. The radius can be found through the ratio of the side of the triangle to the sine of the opposite angle, multiplied by two. For example, R \u003d AE / 2 * sinAME. Similarly, the formula can be written for any of the sides of both triangles.
- Method two: we find the radius of the circumscribed circle through the area of the triangle formed by the diagonal, side and base of the trapezoid: R \u003d AM * ME * AE / 4 * S AME.
Properties of a trapezoid circumscribed about a circle
You can inscribe a circle in a trapezoid if one condition is met. More about it below. And together this combination of figures has a number of interesting properties.
- If a circle is inscribed in a trapezoid, the length of its midline can be easily found by adding the lengths of the sides and dividing the resulting sum in half: m = (c + d)/2.
- For a trapezoid ACME, circumscribed about a circle, the sum of the lengths of the bases is equal to the sum of the lengths of the sides: AK + ME = KM + AE.
- From this property of the bases of a trapezoid, the converse statement follows: a circle can be inscribed in that trapezoid, the sum of the bases of which is equal to the sum of the sides.
- The tangent point of a circle with radius r inscribed in a trapezoid divides the lateral side into two segments, let's call them a and b. The radius of a circle can be calculated using the formula: r = √ab.
- And one more property. In order not to get confused, draw this example yourself. We have the good old ACME trapezoid, circumscribed around a circle. Diagonals are drawn in it, intersecting at the point O. The triangles AOK and EOM formed by the segments of the diagonals and the sides are rectangular.
The heights of these triangles, lowered to the hypotenuses (i.e., the sides of the trapezoid), coincide with the radii of the inscribed circle. And the height of the trapezoid is the same as the diameter of the inscribed circle.
Properties of a rectangular trapezoid
A trapezoid is called rectangular, one of the corners of which is right. And its properties stem from this circumstance.
- A rectangular trapezoid has one of the sides perpendicular to the bases.
- The height and side of the trapezoid adjacent to right angle, are equal. This allows you to calculate the area of a rectangular trapezoid ( general formula S = (a + b) * h/2) not only through the height, but also through the side adjacent to the right angle.
- For a rectangular trapezoid, the general properties of the trapezoid diagonals already described above are relevant.
Proofs of some properties of a trapezoid
Equality of angles at the base of an isosceles trapezoid:
- You probably already guessed that here we again need the ACME trapezoid - draw an isosceles trapezoid. Draw a line MT from vertex M parallel to the side of AK (MT || AK).
The resulting quadrilateral AKMT is a parallelogram (AK || MT, KM || AT). Since ME = KA = MT, ∆ MTE is isosceles and MET = MTE.
AK || MT, therefore MTE = KAE, MET = MTE = KAE.
Where AKM = 180 0 - MET = 180 0 - KAE = KME.
Q.E.D.
Now, based on the property of an isosceles trapezoid (equality of diagonals), we prove that trapezium ACME is isosceles:
- To begin with, let's draw a straight line МХ – МХ || KE. We get a parallelogram KMHE (base - MX || KE and KM || EX).
∆AMH is isosceles, since AM = KE = MX, and MAX = MEA.
MX || KE, KEA = MXE, therefore MAE = MXE.
It turned out that the triangles AKE and EMA are equal to each other, because AM \u003d KE and AE - common side two triangles. And also MAE \u003d MXE. We can conclude that AK = ME, and hence it follows that the trapezoid AKME is isosceles.
Task to repeat
The bases of the trapezoid ACME are 9 cm and 21 cm, the side of the KA, equal to 8 cm, forms an angle of 150 0 with a smaller base. You need to find the area of the trapezoid.
Solution: From vertex K we lower the height to the larger base of the trapezoid. And let's start looking at the angles of the trapezoid.
Angles AEM and KAN are one-sided. Which means they add up to 1800. Therefore, KAN = 30 0 (based on the property of the angles of the trapezoid).
Consider now the rectangular ∆ANK (I think this point is obvious to readers without further proof). From it we find the height of the trapezoid KH - in a triangle it is a leg, which lies opposite the angle of 30 0. Therefore, KN \u003d ½AB \u003d 4 cm.
The area of the trapezoid is found by the formula: S AKME \u003d (KM + AE) * KN / 2 \u003d (9 + 21) * 4/2 \u003d 60 cm 2.
Afterword
If you carefully and thoughtfully studied this article, were not too lazy to draw trapezoids for all the above properties with a pencil in your hands and analyze them in practice, you should have mastered the material well.
Of course, there is a lot of information here, varied and sometimes even confusing: it is not so difficult to confuse the properties of the described trapezoid with the properties of the inscribed one. But you yourself saw that the difference is huge.
Now you have a detailed summary of all common properties trapezoid. As well as specific properties and features of isosceles and rectangular trapezoids. It is very convenient to use to prepare for tests and exams. Try it yourself and share the link with your friends!
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- Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send you important notices and messages.
- We may also use personal information for internal purposes such as auditing, data analysis and various studies to improve the services we provide and to provide you with recommendations regarding our services.
- If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.
Disclosure to third parties
We do not disclose information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.