# Advice 1: How to find the perimeter of an isosceles trapezoid

Trapezoid is a two-dimensional geometric figure with four vertices and only two parallel sides. If the length of its two non-parallel sides equal, then the trapezoid is called isosceles or isosceles. The border of such a polygon is composed of its sides, denoted by the Greek word for "perimeter". Depending on the input data set to compute the perimeter we need to different formulas. Instruction
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If you know the lengths of both bases (a and b) and the length of the sides (c), the perimeter (P) of this geometric shape is calculated very simply. Since the trapezoid isosceles, then its sides have the same length, this means that you are aware of the lengths of all sides - just fold them up: P = a+b+2*c.
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If the lengths of both bases of the trapezoid is unknown, but given the length of the middle line (l) and lateral (c), and these data are sufficient to calculate the perimeter (P). The middle line parallel to both bases and equal in length to their sum. Double this value and add to it, too, twice the length of the sides - this will be the perimeter of an isosceles trapezoid: P = 2*l+2*c.
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If the conditions of the problem are known the lengths of both bases (a and b) and height (h) of an isosceles trapezoid, then using these data it is possible to restore the length of the missing side. This can be done considering a rectangular triangle in which the hypotenuse is the unknown side, and the side - height and short period, which it cuts off from the long base of the trapezoid. The length of this cut can be calculated by dividing in half the difference between the lengths of the larger and smaller bases: (a-b)/2. The length of the hypotenuse (the side of the trapezoid), according to the Pythagorean theorem, is equal to the square root of the sum of squared lengths of the two known sides. Replace in the formula of the first step, the length of the sides of obtained expression, and you get a formula for the perimeter: P = a+b+2*√(h2+(a-b)2/4).
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If the conditions of the problem given length of the smaller base (b) and lateral (c), and also the height of an isosceles trapezoid (h), in considering the same auxiliary triangle, as in the previous step, you will have to calculate the length of the leg. Again, use Pythagorean theorem - the desired value will be equal to the square root of the difference between the squared length of the side (hypotenuse) and height (side): √(c2-h2). On this segment of the unknown base of the trapezoid, you can restore its length double this expression and add to the result the length of the short base: b+2*√(c2-h2). Substitute this expression into the formula from step one and find the perimeter of an isosceles trapezoid: P = b+2*√(c2-h2)+b+2*c = 2*(√(c2-h2)+b+c).

# Advice 2 : How to find the perimeter of a rectangular trapezoid

Trapezoid - a quadrilateral with two parallel bases and non-parallel sides. A rectangular trapezoid has a right angle at one side. Instruction
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The perimeter of a rectangular trapezoid equal to the sum of the lengths of the sides of the two bases and two sides. Task 1. Find the perimeter of a rectangular trapezoid, if the lengths of all its sides. To do this, add up all four values: P (perimeter) = a + b + c + d.This is the easiest way to find the perimeter problems, other initial data, in the end, it boils down to it. Let's consider the options.
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Task 2.Find the perimeter of a rectangular trapezoid, if you know the bottom base AD = a, not perpendicular to it side CD = d, and the angle at this side of the ADC is equal to alpha.Solution.Guide the height of the trapezoid from vertex C is on the larger base, get the CE segment, trapezoid split into two pieces - a rectangle ABCE and right triangle ECD. The hypotenuse of the triangle is known to us side of the trapezoid CD, one of the legs is equal to is perpendicular to the side of the trapezoid (rule rectangle two parallel sides equal to AB = CE), and another segment whose length is equal to the difference between the bases of the trapezoid ED = AD - BC.
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Find the sides of a triangle: according to the existing formula CE = CD*sin(ADC) and ED = CD*cos(ADC).Now calculate the upper base - BC = AD - ED = a - CD*cos(ADC) = a - d*cos(alpha).Find out the length of the perpendicular sides AB = CE = d*sin(alpha).So you've got the lengths of all sides of a rectangular trapezoid.
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Fold the values obtained, it will be a rectangular perimeter of a trapezoid:P = AB + BC + CD + AD = d*sin(alpha) + (a - d*cos(alpha)) + d + a = 2*a + d*(sin(alpha) cos(alpha) + 1).
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Task 3.Find the perimeter of a rectangular trapezoid, if the lengths of its bases AD = a, BC = c, the length of the perpendicular sides AB = b and the acute angle at the other side ADC = the alpha.Solution.Spend the perpendicular CE will receive a rectangle ABCE and a triangle CED.Now find the length of the hypotenuse of the triangle CD = AB/sin(ADC) = b/sin(alpha).So you've got the lengths of all sides.
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Fold the resulting values:P = AB + BC + CD + AD = b + c + b/sin(alpha) + a = a + b*(1+1/sin(alpha) + C.

# Advice 3 : How to find side of a trapezoid

A-line is a regular quadrilateral, having the additional property of parallelism of its two sides, called bases. So the question, first, it should be understood from the point of view of finding the sides. Secondly, for the job of a trapezoid requires at least four parameters. Instruction
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In this particular case, the General reference (not excessive) should be considered as the condition: the length of the upper and lower bases, and the vector of one of the diagonals. Indices of the coordinates (in order writing formulas was not like multiplication) are shown in italics).For the graphic image of the solution process build figure 1.
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Let in the problem is considered a-line AВCD. It contains the lengths of the bases of the armed forces=b and AD=a and the diagonal AC, given as a vector p(px, py). Its length (modulus) |p|=p=sqrt(((px)^2 +(py)^2). Since the vector is set and an angle of inclination to the axis (task - 0X), then label it using f (angle CAD and parallel the angle ACB). Next, you need to apply known to the theorem of cosines. This desired value (the length of a CD or AV when setting up the equation denote by x).
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Consider the triangle AСD. Here, the length of side AC is equal to the module of the vector |p|=p. AD=b. By theorem of cosines x^2=p^2+ b^2-2pbcosф. x=CD=sqrt(p^2+ b^2-2pbcosф)=CD.
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Now consider the triangle ABC. Length of side AC is equal to the module of the vector |p|=p. BC=a. By theorem of cosines x^2=p^2+ a^2-2расоѕф. x=AB=sqrt(p^2+ a^2-2расоѕф).
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Although the quadratic equation has two roots, in this case, you should choose only those where the root of the discriminant of a plus sign, while deliberately excluding negative decisions. This is because the length of the sides of the trapezoid must obviously be positive.
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Thus, the required solutions in the form of algorithms solving this problem are obtained. To represent numeric, the decision is to substitute data from the conditions. While Sof is calculated as the direction vector (ORT) vector p=px/sqrt(px^2+py^2).
Note
Of course, other initial data, for example, the specification of the two diagonals and the height of the trapezoid. But in any case you need information about the distance between the bases of the trapezoid.
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