Project work “Interesting properties of a trapezoid” Completed by: 10th grade students Kudzaeva Ellina Bazzaeva Diana MCOU Secondary School s. N.Batako Head: Gagieva A.O. November 20, 2015
Purpose of the work: To consider the properties of the trapezoid, which are not studied in the school geometry course, but when solving geometric problems of the Unified State Exam from the expanded part C 4, it may be necessary to know and be able to apply precisely these properties.
Properties of a trapezoid: If a trapezoid is divided by a line parallel to its bases equal to a and b, into two equal trapezoids. Then the segment to this line, enclosed between the lateral sides, is equal to a B to
Property of a segment passing through the point of intersection of the diagonals of a trapezoid. The segment parallel to the bases passing through the point of intersection of the diagonals is equal to: a in c
Properties of a trapezoid: A straight line segment parallel to the bases of a trapezoid, enclosed inside the trapezoid, is divided into three parts by its diagonals. Then the segments adjacent to the sides are equal to each other. MP=OK R M O K
Properties of an isosceles trapezoid: If a circle can be inscribed in a trapezoid, then the radius of the circle is the average proportional to the segments into which the tangent point divides the side. O S V A D. E O
Properties of an isosceles trapezoid: If the center of the circumscribed circle lies at the base of the trapezoid, then its diagonal is perpendicular to the side O A B C D
Properties of an isosceles trapezoid: A circle can be inscribed in an isosceles trapezoid if the side side is equal to its midline. S V A D h
1) If the problem statement says that a circle is inscribed in a rectangular trapezoid, you can use the following properties: 1. The sum of the bases of the trapezoid is equal to the sum of the sides. 2. The distances from the vertex of the trapezoid to the tangent points of the inscribed circle are equal. 3. The height of a rectangular trapezoid is equal to its smaller side and equal to the diameter of the inscribed circle. 4. The center of the inscribed circle is the point of intersection of the bisectors of the angles of the trapezoid. 5. If the tangent point divides the side into segments m and n, then the radius of the inscribed circle is equal to
Properties of a rectangular trapezoid into which a circle is inscribed: 1) A quadrilateral formed by the center of the inscribed circle, the points of contact and the vertex of the trapezoid - a square whose side is equal to the radius. (AMOE and BKOM are squares with side r). 2) If a circle is inscribed in a rectangular trapezoid, then the area of the trapezoid is equal to the product of its bases: S=AD*BC
Proof: The area of a trapezoid is equal to the product of half the sum of its bases and its height: Let us denote CF=m, FD=n. Since the distances from the vertices to the tangent points are equal, the height of the trapezoid is equal to two radii of the inscribed circle, and
I. The bisectors of the angles at the lateral side of the trapezoid intersect at an angle of 90º. 1)∠ABC+∠BAD=180º (as internal one-sided with AD∥BC and secant AB). 2) ∠ABK+∠KAB=(∠ABC+∠BAD):2=90º (since the bisectors bisect the angles). 3) Since the sum of the angles of a triangle is 180º, in triangle ABK we have: ∠ABK+∠KAB+∠AKB=180º, hence ∠AKB=180-90=90º. Conclusion: The angle bisectors on the lateral side of a trapezoid intersect at right angles. This statement is used when solving problems on a trapezoid into which a circle is inscribed.
I I. The point of intersection of the bisectors of the trapezoid adjacent to the lateral side lies on the midline of the trapezoid. Let the bisector of angle ABC intersect side AD at point S. Then triangle ABS is isosceles with base BS. This means that its bisector AK is also a median, that is, point K is the midpoint of BS. If M and N are the midpoints of the lateral sides of the trapezoid, then MN is the midline of the trapezoid and MN∥AD. Since M and K are the midpoints of AB and BS, then MK is the midline of the triangle ABS and MK∥AS. Since only one line parallel to this one can be drawn through point M, point K lies on the midline of the trapezoid.
III. The point of intersection of the bisectors of acute angles at the base of a trapezoid belongs to another base. In this case, triangles ABK and DCK are isosceles with bases AK and DK, respectively. Thus, BC=BK+KC=AB+CD. Conclusion: If the bisectors of the acute angles of a trapezoid intersect at a point belonging to the smaller base, then the smaller base is equal to the sum of the lateral sides of the trapezoid. An isosceles trapezoid in this case has a smaller base twice the size of its side.
I V. The point of intersection of the bisectors of obtuse angles at the base of the trapezoid belongs to another base. In this case, triangles ABF and DCF are isosceles with bases BF and CF, respectively. Hence AD=AF+FD=AB+CD. Conclusion: If the bisectors of the obtuse angles of a trapezoid intersect at a point belonging to the larger base, then the larger base is equal to the sum of the lateral sides of the trapezoid. In this case, an isosceles trapezoid has a larger base that is twice as large as its side.
If an isosceles trapezoid with sides a, b, c, d can be inscribed and circles can be drawn around it, then the area of the trapezoid is
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- (Greek trapezion). 1) in geometry, a quadrilateral in which two sides are parallel and two are not. 2) a figure adapted for gymnastic exercises. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. TRAPEZE... ... Dictionary of foreign words of the Russian language
Trapezoid- Trapezoid. TRAPEZE (from the Greek trapezion, literally table), a convex quadrilateral in which two sides are parallel (the bases of the trapezoid). The area of a trapezoid is equal to the product of half the sum of the bases (midline) and the height. ... Illustrated Encyclopedic Dictionary
Quadrangle, projectile, crossbar Dictionary of Russian synonyms. trapezoid noun, number of synonyms: 3 crossbar (21) ... Synonym dictionary
- (from the Greek trapezion, literally table), a convex quadrangle in which two sides are parallel (the bases of a trapezoid). The area of a trapezoid is equal to the product of half the sum of the bases (midline) and the height... Modern encyclopedia
- (from the Greek trapezion, lit. table), a quadrilateral in which two opposite sides, called the bases of the trapezoid, are parallel (in the figure AD and BC), and the other two are non-parallel. The distance between the bases is called the height of the trapezoid (at ... ... Big Encyclopedic Dictionary
TRAPEZOUS, a quadrangular flat figure in which two opposite sides are parallel. The area of a trapezoid is equal to half the sum of the parallel sides multiplied by the length of the perpendicular between them... Scientific and technical encyclopedic dictionary
TRAPEZE, trapezoid, women's (from Greek trapeza table). 1. Quadrilateral with two parallel and two non-parallel sides (mat.). 2. A gymnastic apparatus consisting of a crossbar suspended on two ropes (sports). Acrobatic... ... Ushakov's Explanatory Dictionary
TRAPEZE, and, female. 1. A quadrilateral with two parallel and two non-parallel sides. The bases of the trapezoid (its parallel sides). 2. A circus or gymnastics apparatus is a crossbar suspended on two cables. Ozhegov's explanatory dictionary. WITH … Ozhegov's Explanatory Dictionary
Female, geom. a quadrilateral with unequal sides, two of which are parallel (parallel). Trapezoid, a similar quadrilateral in which all sides run apart. Trapezohedron, a body faceted by trapezoids. Dahl's Explanatory Dictionary. IN AND. Dahl. 1863 1866 … Dahl's Explanatory Dictionary
- (Trapeze), USA, 1956, 105 min. Melodrama. Aspiring acrobat Tino Orsini joins a circus troupe where Mike Ribble, a famous former trapeze artist, works. Mike once performed with Tino's father. Young Orsini wants Mike... Encyclopedia of Cinema
A quadrilateral in which two sides are parallel and the other two sides are not parallel. The distance between parallel sides is called. height T. If parallel sides and height contain a, b and h meters, then the area of T contains square meters ... Encyclopedia of Brockhaus and Efron
In this article we will try to reflect the properties of a trapezoid as fully as possible. In particular, we will talk about the general characteristics and properties of a trapezoid, as well as the properties of an inscribed trapezoid and 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 properties discussed will help you sort it into places in your head and better remember the material.
Trapeze and all-all-all
To begin with, let us briefly recall what a trapezoid is and what other concepts are associated with it.
So, a trapezoid is a quadrilateral figure, two of whose sides are parallel to each other (these are the bases). And the two are not parallel - these are the sides.
In a trapezoid, the height can be lowered - perpendicular to the bases. The center line and diagonals are drawn. It is also possible to draw a bisector from any angle of the trapezoid.
We will now talk about the various properties associated with all these elements and their combinations.
Properties of trapezoid diagonals
To make it clearer, while you are reading, sketch out the trapezoid ACME 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 HT lies on the midline. And its length can be obtained by dividing the difference of the bases by two: ХТ = (a – b)/2.
- Before us is the same trapezoid ACME. The diagonals intersect at point O. Let's look at the triangles AOE and MOK, formed by segments of the diagonals together with the bases of the trapezoid. These triangles are similar. The similarity coefficient k of triangles is expressed through the ratio of the bases of the trapezoid: k = AE/KM.
The ratio of the areas of triangles AOE and MOK is described by the coefficient k 2 . - The same trapezoid, the same diagonals intersecting at point O. Only this time we will consider the triangles that the segments of the diagonals formed together with the sides of the trapezoid. The areas of triangles AKO and EMO are equal in size - their areas are the same.
- Another property of a trapezoid involves the construction of diagonals. So, if you continue the sides of AK and ME in the direction of the smaller base, then sooner or later they will intersect at a certain point. Next, draw a straight line through the middle 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 connect together the point of intersection of the diagonals of the trapezoid O, the point at which the extensions of the sides and the middle of the bases X and T intersect. - Through the point of intersection of the diagonals we will draw a segment that will connect the bases of the trapezoid (T lies on the smaller base KM, X on the larger AE). The intersection point of the diagonals divides this segment in the following ratio: TO/OX = KM/AE.
- Now, through the point of intersection of the diagonals, we will 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 the segment using the formula 2ab/(a + b).
Properties of the midline of a trapezoid
Draw the middle line in the trapezoid 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.
Trapezoid bisector property
Select any angle of the trapezoid and draw a bisector. Let's take, for example, the angle KAE of our trapezoid ACME. Having completed the construction yourself, you can easily verify 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.
Properties of trapezoid angles
- Whichever of the two pairs of angles adjacent to the side you choose, the sum of the angles in the pair is always 180 0: α + β = 180 0 and γ + δ = 180 0.
- Let's 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 segment TX can be easily calculated based on the difference in the lengths of the bases, divided in half: TX = (AE – KM)/2.
- If parallel lines are drawn through the sides of a trapezoid angle, they will divide the sides of the angle into proportional segments.
Properties of an isosceles (equilateral) trapezoid
- In an isosceles trapezoid, the angles at any base are equal.
- Now build a trapezoid again to make it easier to imagine what we're talking about. Look carefully at the base AE - the vertex of the opposite base 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 middle line of the 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 around an isosceles trapezoid can a circle be described, since the sum of the opposite angles of a quadrilateral is 180 0 - a prerequisite for this.
- The property of an isosceles trapezoid follows from the previous paragraph - if a circle can be described near the trapezoid, it is isosceles.
- From the features of an isosceles trapezoid follows the property of the height of a trapezoid: if its diagonals intersect at right angles, then the length of the height is equal to half the sum of the bases: h = (a + b)/2.
- Again, draw the segment TX 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 the height from the opposite vertex of the trapezoid onto the larger base (let's call it a). You will get two segments. 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 us dwell on this issue in more detail. In particular, on where the center of the circle is in relation to the trapezoid. Here, too, it is recommended that you take the time to pick up a pencil and draw what will be discussed below. This way you will understand faster and remember better.
- The location of the center of the circle is determined by the angle of inclination of the trapezoid's diagonal to its side. For example, a diagonal may extend from the top of a trapezoid at right angles to the side. In this case, the larger base intersects the center of the circumcircle 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 trapezoid, beyond its larger base, if there is an obtuse angle between the diagonal of the trapezoid and the side.
- The angle formed by the diagonal and the large base of the trapezoid ACME (inscribed angle) is half the central angle that corresponds to it: MAE = ½MOE.
- Briefly about two ways to find the radius of a circumscribed circle. Method one: look carefully at your drawing - what do you see? You can easily notice that the diagonal splits the trapezoid into two triangles. The radius can be found by the ratio of the side of the triangle to the sine of the opposite angle, multiplied by two. For example, R = AE/2*sinAME. In a similar way, the formula can be written for any of the sides of both triangles.
- Method two: find the radius of the circumscribed circle through the area of the triangle formed by the diagonal, side and base of the trapezoid: R = AM*ME*AE/4*S AME.
Properties of a trapezoid circumscribed about a circle
You can fit a circle into a trapezoid if one condition is met. Read 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 the trapezoid ACME, described 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 a trapezoid whose sum of bases is equal to the sum of its sides.
- The tangent point of a circle with radius r inscribed in a trapezoid divides the 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. To avoid confusion, draw this example yourself too. We have the good old trapezoid ACME, described around a circle. It contains diagonals that intersect at point O. The triangles AOK and EOM formed by the segments of the diagonals and the lateral sides are rectangular.
The heights of these triangles, lowered to the hypotenuses (i.e., the lateral sides of the trapezoid), coincide with the radii of the inscribed circle. And the height of the trapezoid coincides with the diameter of the inscribed circle.
Properties of a rectangular trapezoid
A trapezoid is called rectangular if one of its angles is right. And its properties stem from this circumstance.
- A rectangular trapezoid has one of its sides perpendicular to its base.
- The height and side of a trapezoid adjacent to a 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 diagonals of a trapezoid already described above are relevant.
Evidence of some properties of the trapezoid
Equality of angles at the base of an isosceles trapezoid:
- You probably already guessed that here we will need the AKME trapezoid again - draw an isosceles trapezoid. Draw a straight 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 does 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 trapezoid ACME is isosceles:
- First, let’s draw a straight line MX – MX || KE. We obtain a parallelogram KMHE (base – MX || KE and KM || EX).
∆AMX is isosceles, since AM = KE = MX, and MAX = MEA.
MH || KE, KEA = MXE, therefore MAE = MXE.
It turned out that the triangles AKE and EMA are equal to each other, since AM = KE and AE are the common side of the two triangles. And also MAE = MXE. We can conclude that AK = ME, and from this it follows that the trapezoid AKME is isosceles.
Review task
The bases of the trapezoid ACME are 9 cm and 21 cm, the side side KA, equal to 8 cm, forms an angle of 150 0 with the 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. This means that in total they give 180 0. Therefore, KAN = 30 0 (based on the property of trapezoidal angles).
Let us now consider the rectangular ∆ANC (I believe this point is obvious to readers without additional evidence). From it we will find the height of the trapezoid KH - in a triangle it is a leg that lies opposite the angle of 30 0. Therefore, KH = ½AB = 4 cm.
We find the area of the trapezoid using the formula: S ACME = (KM + AE) * KN/2 = (9 + 21) * 4/2 = 60 cm 2.
Afterword
If you carefully and thoughtfully studied this article, were not too lazy to draw trapezoids for all the given 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 have seen that the difference is huge.
Now you have a detailed outline of all the general properties of a trapezoid. As well as specific properties and characteristics 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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