In this article, we will try to reflect the properties of the trapezoid as fully as possible. In particular, we will talk about the general signs 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.
About the various properties associated with all these elements and their combinations, we will now talk.
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 connects 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 of the central angle that 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 the 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 is the common side of the 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 lateral 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 the general properties of a 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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- The segment connecting the midpoints of the diagonals of a trapezoid is equal to half the difference of the bases
- The triangles formed by the bases of the trapezoid and the segments of the diagonals up to the point of their intersection are similar
- Triangles formed by segments of the diagonals of a trapezoid, the sides of which lie on the sides of the trapezoid - equal area (have the same area)
- If we extend the sides of the trapezoid towards the smaller base, then they will intersect at one point with the straight line connecting the midpoints of the bases
- The segment connecting the bases of the trapezoid, and passing through the point of intersection of the diagonals of the trapezoid, is divided by this point in a proportion equal to the ratio of the lengths of the bases of the trapezoid
- A segment parallel to the bases of the trapezoid and drawn through the intersection point of the diagonals is bisected by this point, and its length is equal to 2ab / (a + b), where a and b are the bases of the trapezoid
Properties of a segment connecting the midpoints of the diagonals of a trapezoid
Connect the midpoints of the diagonals of the trapezoid ABCD, as a result of which we will have a segment LM.
A line segment that joins the midpoints of the diagonals of a trapezoid lies on the midline of the trapezium.
This segment parallel to the bases of the trapezium.
The length of the segment connecting the midpoints of the diagonals of a trapezoid is equal to the half-difference of its bases.
LM = (AD - BC)/2
or
LM = (a-b)/2
Properties of triangles formed by the diagonals of a trapezoid
The triangles that are formed by the bases of the trapezoid and the point of intersection of the diagonals of the trapezoid - are similar.
Triangles BOC and AOD are similar. Because the angles BOC and AOD are vertical, they are equal.
Angles OCB and OAD are internal crosswise lying at parallel lines AD and BC (the bases of the trapezium are parallel to each other) and the secant line AC, therefore, they are equal.
Angles OBC and ODA are equal for the same reason (internal cross-lying).
Since all three angles of one triangle are equal to the corresponding angles of another triangle, these triangles are similar.
What follows from this?
To solve problems in geometry, the similarity of triangles is used as follows. If we know the lengths of the two corresponding elements of similar triangles, then we find the similarity coefficient (we divide one by the other). From where the lengths of all other elements are related to each other by exactly the same value.
Properties of triangles lying on the lateral side and diagonals of a trapezoid
Consider two triangles lying on the sides of the trapezoid AB and CD. These are triangles AOB and COD. Despite the fact that the sizes of individual sides of these triangles can be completely different, but the areas of the triangles formed by the sides and the point of intersection of the diagonals of the trapezoid are, that is, the triangles are equal.
If the sides of the trapezoid are extended towards the smaller base, then the point of intersection of the sides will be coincide with a straight line that passes through the midpoints of the bases.
Thus, any trapezoid can be extended to a triangle. Wherein:
- The triangles formed by the bases of a trapezoid with a common vertex at the point of intersection of the extended sides are similar
- The straight line connecting the midpoints of the bases of the trapezoid is, at the same time, the median of the constructed triangle
Properties of a segment connecting the bases of a trapezoid
If you draw a segment whose ends lie on the bases of the trapezoid, which lies at the intersection point of the diagonals of the trapezoid (KN), then the ratio of its constituent segments from the side of the base to the intersection point of the diagonals (KO / ON) will be equal to the ratio of the bases of the trapezoid(BC/AD).
KO/ON=BC/AD
This property follows from the similarity of the corresponding triangles (see above).
Properties of a segment parallel to the bases of a trapezoid
If you draw a segment parallel to the bases of the trapezoid and passing through the intersection point of the diagonals of the trapezoid, then it will have the following properties:
- Preset distance (KM) bisects the point of intersection of the diagonals of the trapezoid
- Cut length, passing through the point of intersection of the diagonals of the trapezoid and parallel to the bases, is equal to KM = 2ab/(a + b)
Formulas for finding the diagonals of a trapezoid
a, b- bases of a trapezoid
c, d- sides of the trapezoid
d1 d2- diagonals of a trapezoid
α β - angles with a larger base of the trapezoid
Formulas for finding the diagonals of a trapezoid through the bases, sides and angles at the base
The first group of formulas (1-3) reflects one of the main properties of the trapezoid diagonals:
1. The sum of the squares of the diagonals of a trapezoid is equal to the sum of the squares of the sides plus twice the product of its bases. This property of the diagonals of a trapezoid can be proved as a separate theorem
2 . This formula is obtained by transforming the previous formula. The square of the second diagonal is thrown over the equal sign, after which the square root is extracted from the left and right sides of the expression.
3 . This formula for finding the length of the diagonal of a trapezoid is similar to the previous one, with the difference that another diagonal is left on the left side of the expression
The next group of formulas (4-5) is similar in meaning and expresses a similar relationship.
The group of formulas (6-7) allows you to find the diagonal of a trapezoid if you know the larger base of the trapezoid, one side and the angle at the base.
Formulas for finding the diagonals of a trapezoid in terms of height
Task.
The diagonals of the trapezoid ABCD (AD | | BC) intersect at point O. Find the length of the base BC of the trapezoid if the base AD = 24 cm, length AO = 9 cm, length OS = 6 cm.
Decision.
The solution of this task is absolutely identical to the previous tasks in terms of ideology.
Triangles AOD and BOC are similar in three angles - AOD and BOC are vertical, and the remaining angles are pairwise equal, since they are formed by the intersection of one line and two parallel lines.
Since the triangles are similar, then all their geometric dimensions are related to each other, as the geometric dimensions of the segments AO and OC known to us by the condition of the problem. I.e
AO/OC=AD/BC
9 / 6 = 24 / B.C.
BC = 24 * 6 / 9 = 16
Answer: 16 cm
Task .
In the trapezoid ABCD it is known that AD=24, BC=8, AC=13, BD=5√17. Find the area of the trapezoid. 
Decision .
To find the height of a trapezoid from the vertices of the smaller base B and C, we lower two heights onto the larger base. Since the trapezoid is unequal, we denote the length AM = a, the length KD = b ( not to be confused with the symbols in the formula finding the area of a trapezoid). Since the bases of the trapezoid are parallel and we have omitted two heights perpendicular to the larger base, then MBCK is a rectangle.
Means
AD=AM+BC+KD
a + 8 + b = 24
a = 16 - b
Triangles DBM and ACK are right-angled, so their right angles are formed by the heights of the trapezoid. Let's denote the height of the trapezoid as h. Then by the Pythagorean theorem
H 2 + (24 - a) 2 \u003d (5√17) 2
and
h 2 + (24 - b) 2 \u003d 13 2
Consider that a \u003d 16 - b, then in the first equation
h 2 + (24 - 16 + b) 2 \u003d 425
h 2 \u003d 425 - (8 + b) 2
Substitute the value of the square of the height into the second equation, obtained by the Pythagorean Theorem. We get:
425 - (8 + b) 2 + (24 - b) 2 = 169
-(64 + 16b + b) 2 + (24 - b) 2 = -256
-64 - 16b - b 2 + 576 - 48b + b 2 = -256
-64b = -768
b = 12
Thus, KD = 12
Where
h 2 \u003d 425 - (8 + b) 2 \u003d 425 - (8 + 12) 2 \u003d 25
h = 5
Find the area of a trapezoid using its height and half the sum of the bases
, where a b - the bases of the trapezoid, h - the height of the trapezoid
S \u003d (24 + 8) * 5 / 2 \u003d 80 cm 2
Answer: the area of a trapezoid is 80 cm2.
Trapeze is a quadrilateral with two parallel sides, which are the bases, and two non-parallel sides, which are the sides.
There are also names such as isosceles or isosceles.
It is a trapezoid with right angles on the lateral side.
Trapeze elements
a, b bases of a trapezoid(a parallel to b ),
m, n — sides trapeze,
d 1 , d 2 — diagonals trapeze,
h- height trapezoid (a segment connecting the bases and at the same time perpendicular to them),
MN- middle line(a segment connecting the midpoints of the sides).
Trapezium area
- Through half the sum of the bases a, b and the height h : S = \frac(a + b)(2)\cdot h
- Through the midline MN and height h : S = MN\cdot h
- Through the diagonals d 1 , d 2 and the angle (\sin \varphi ) between them: S = \frac(d_(1) d_(2) \sin \varphi)(2)
Trapezoid Properties
Median line of the trapezoid
middle line is parallel to the bases, equal to their half-sum, and divides each segment with ends located on straight lines that contain the bases (for example, the height of the figure) in half:
MN || a, MN || b, MN = \frac(a + b)(2)
The sum of the angles of a trapezoid
The sum of the angles of a trapezoid, adjacent to each side, is equal to 180^(\circ) :
\alpha + \beta = 180^(\circ)
\gamma + \delta =180^(\circ)
Equal area triangles of a trapezoid
Equal-sized, that is, having equal areas, are the segments of the diagonals and the triangles AOB and DOC formed by the sides.
Similarity of formed trapezoid triangles
similar triangles are AOD and COB, which are formed by their bases and diagonal segments.
\triangle AOD \sim \triangle COB
similarity coefficient k is found by the formula:
k = \frac(AD)(BC)
Moreover, the ratio of the areas of these triangles is equal to k^(2) .
The ratio of the lengths of segments and bases
Each segment connecting the bases and passing through the point of intersection of the diagonals of the trapezoid is divided by this point in relation to:
\frac(OX)(OY) = \frac(BC)(AD)
This will also be true for the height with the diagonals themselves.
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 known in ancient Greece., which was described in more detail by Aristotle in his scientific work "Beginnings". And the knowledge that was developed thousands of years ago is still relevant today. Therefore, we will get acquainted with them in more detail.
In contact with
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 of geometric shapes: 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, 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, in simple terms, 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.
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
Decision. 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, a more rational solution and search for the required value is possible.
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 a 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;
- square;
- 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
Conclusion
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.
