Each trapezoid has two sides and two bases. In order to find out the area, perimeter or other parameters of this figure, you need to know at least one of the sides. Also, often, according to the conditions of the problem, it is required to find the side side of a rectangular trapezoid.

Instructions

Draw a rectangular trapezoid ABCD. Label the sides of this figure as AB and DC, respectively. The first lateral side of DC coincides with the height of the trapezoid. It is perpendicular to the two bases of a rectangular trapezoid.
There are several ways to find the sides. So, for example, if the problem is given the second side BA and the angle ABH = 60, then find the first height in the simplest way by finding the height BH:
BH=AB*sin?
Since BH=CD, then CD=AB*sin?=?3AB/2

If, on the contrary, you are given a side of a trapezoid, designated as CD, and you need to find its side AB, this problem is solved in a slightly different way. Since BH=CD, and at the same time, BH represents the leg of the triangle ABH, we can conclude that the side AB is equal to:
AB=BH/sin?=2BH/?3

The problem can also be solved if the values ​​of the angles are unknown, provided that two bases and a side AB are given. However, in this case, you can only find side CD, which is the height of the trapezoid. Initially, knowing the values ​​of the bases, find the length of the segment AH. It is equal to the difference between the greater and lesser bases, since it is known that BH = CD:
AH=AD-BC
Then, using the Pythagorean theorem, find the height of BH, equal to the side CD:
BH=?AB^2-AH^2

If a rectangular trapezoid has a diagonal BD and an angle 2?, as shown in Figure 2, then the side AB can also be found using the Pythagorean theorem. To do this, first calculate the length of the base AD:
AD=BD*cos2?
Then find side AB like this:
AB=?BD^2-AD^2
After this, prove the similarity of triangles ABD and BCD. Since these triangles have one common side - the diagonal, and at the same time, two angles are equal, as can be seen from the figure, then these figures are similar. Based on this evidence, find the second side. If the upper base and diagonal are known, then find the side in the usual way using the standard cosine theorem:
c^2=a^2+b^2-2ab cos?, where a, b, c are the sides of the triangle, ? - the angle between sides a and b.

A trapezoid is an ordinary quadrilateral that has the additional property of parallelism of its two sides, which are called bases. Therefore, this question, firstly, should be understood from the point of view of finding the sides. Secondly, for the task trapezoids At least four parameters are required.

Instructions

In this particular case, its most general task (not redundant) should be considered the condition: the lengths of the upper and lower bases, as well as the vector of one of the diagonals, are given. Coordinate indices (so that the writing does not resemble multiplication) will be in italics). To graphically depict the solution process, draw Figure 1.

Let the presented problem consider the trapezoid ABCD. It gives the lengths of the bases BC=b and AD=a, as well as the diagonal AC, specified by the vector p(px, py). Its length (modulus) |p|=p=sqrt(((px)^2 +(py)^2). Since the vector is also given by the angle of inclination to the axis (in the problem - 0X), then denote it by φ ( angle CAD and angle ACB parallel to it). Next, you need to apply the cosine theorem, known from the school curriculum. In this case, denote the lengths CD or AB by x.

Now consider triangle ABC. Length sides AC is equal to the modulus of the vector |p|=p. BC=a. By the cosine theorem x^2=p^2+ a^2-2pacosф. x=AB=sqrt(p^2+ a^2-2pacosф).

Although quadratic equation and has two roots, in in this case it is necessary to select only those where there is a plus sign in front of the root of the discriminant, while deliberately excluding negative solutions. This is due to the fact that the length sides trapezoids must be obviously positive.

So, the required solutions in the form of algorithms for solving this problem have been obtained. To present a numerical solution, all that remains is to substitute the data from the condition. In this case, cosф is calculated as the direction vector (ort) of the vector p=px/sqrt(px^2+py^2).

note

Of course, other initial data are also possible, for example, specifying two diagonals and the height of the trapezoid. But in any case, you will need information about the distance between the bases of the trapezoid.

A trapezoid is a geometric figure with four corners, two sides of which are parallel to each other and are called bases, and the other two are not parallel and are called lateral.

Instructions

Let's consider two problems with different initial data. Problem 1. Find the side side isosceles trapezoids, if known base BC = b, base AD = d and lateral angle BAD = Alpha. Solution: Lower the perpendicular (height trapezoids) from vertex B to the intersection with the big one base m, get the segment BE. Write AB using the formula in terms of the angle: AB = AE/cos(BAD) = AE/cos(Alpha).

Find AE. It will be equal to the difference in the lengths of the two bases, divided in half. So: AE = (AD - BC)/2 = (d - b)/2.Now find AB = (d - b)/(2*cos(Alpha)).In the isosceles trapezoids the lengths of the sides are equal, therefore, CD = AB = (d - b)/(2*cos(Alpha)).

Task 2. Find the side side trapezoids AB, if the upper one is known base BC = b- bottom base AD = d- height BE = h and the angle at the opposite side CDA is equal to Alpha. Solution: Draw the second height from vertex C to the intersection with the bottom base m, get segment CF. Consider the right triangle CDF, find side FD using the following formula: FD = CD*cos(CDA). Find the length of the side of CD from another formula: CD = CF/sin(CDA). So: FD = CF*cos(CDA)/sin(CDA). CF = BE = h, therefore FD = h*cos(Alpha)/sin(Alpha) = h*ctg(Alpha).

Consider the right triangle ABE. Knowing the lengths of its sides AE and BE, you can find the third side- hypotenuse AB. You know the length of side BE, find AE as follows: AE = AD - BC - FD = d - b - h*ctg(Alpha).Using next property right triangle - the square of the hypotenuse is equal to the sum of the squares of the legs - find AB:AB(2) = h(2) + (d - b - h*ctg(Alpha))(2).The value of the side trapezoids AB is equal square root from the expression located in right side equality.

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Geometry is a science that begins to be studied in school. It is a mistake to think that it will not be useful in life. Sometimes exact dimensions of figures are needed to make, for example, a WEB design for a room. And there are different shapes, including trapezoids. Often you need to find the values ​​of their sides or bases. Let's look in detail at how to find the lateral side of a given quadrilateral of various shapes, if its angles, bases, diagonals, area, etc. are known.

How to find the side of a trapezoid if the bases are known?

A trapezoid is a quadrilateral with only two parallel sides. And these non-intersecting segments are called the bases of this figure. Trapezes come in different varieties:

• Isolaterals are those whose sides are equal.
• Rectangular - have one right angle at the base.
• Acute-angled, versatile - with two acute angles at the base.
• Obtuse, scalene - with one obtuse angle at the base.

Consider the option of finding the side (height) of a rectangular trapezoid if you are given the values ​​of the bases.

To solve this problem, you will need to do the following:

• Draw the second height - BH in the quadrilateral.
• The resulting segment BN = SD, since the base BC is parallel to AD.
• The resulting triangle ABC is isosceles, because AC is a bisector, respectively, the angles at the base are equal and AB = CB = 10 cm.
• Let's consider the triangle ABN; in fact, we know its two sides: BA and AN. AN = BP - CD = 16 - 10 = 6 cm.
• Hence, according to the Pythagorean theorem: ВН² = AB² - HA² = 64; VN = 8 cm, respectively, SD is also equal to 8 centimeters.

In addition, if you know the angle VAD, then SD = (AD - BC) tan α or SD = AB sin α.

The large lateral side is calculated using the following formulas:

• AB² = CD² + (AD - BC)²
• AB = (AD - BC)/cos ∠BAN
• AB = CD/sin ∠VAN

How to find the side side of a rectangular trapezoid if the diagonals, area, and midline are known?

If we denote the height of the trapezoid - b, the larger side - c, the base - a and k, the diagonals - d1 and d2. The larger angle between them is β, the smaller is α, then the height (side of the trapezoid) can be found using the following formulas:

b = d2 d1/ (a + k) sin α;

or b = d2 d1/ (a + k) sin β

In order to determine b - the smaller side of a rectangular trapezoid, c - the larger side of the figure, with known data S - area, n - midline, use the following calculations:

b = S/n = 2S/ (a + k)

c = S/n sin α = 2S/ (a + k) sin α

How to find the sides of an isosceles trapezoid?

So, for an isosceles trapezoid, AB = DC. If you are given different values, then the sides can be found using the formulas below:

• if the height - h and angle - α are known, then AB = DC = h/ sin α;
• if the values ​​of the bases and the angle are given - α, then AB = DC = (a - b) / cos α;
• if diagonals d and bases are given, then AB² = DC² = d² - b a;
• if the values ​​are known midline- l, area - S, angles - α or - β (at the top near the base b, then AB = DC = S/ l sin α = S/ l sin α.

AB = DC = S/ (b + a) sin α = S/ (b + a) sin β

In the future, if you learn the formulas and learn to correctly draw drawings of these figures, then solving a geometry problem will not be difficult for you. After all, with the right picture, the answer to the problem is almost immediately visible.