You will need

- - all side of the trapezoid (AB, BC, CD, DA).

Instruction

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Non-parallel

**sides****of a trapezoid**are called lateral sides, and parallel bases. Line between the bases, perpendicular to them - the height**of the trapezoid**. If the lateral**sides****of a trapezoid**are equal, then it is called isosceles. First, consider the solution**of a trapezoid**that is not isosceles.2

Guide BE cut from point B to lower the base AD parallel to the side

**of the trapezoid**CD. Since BE and CD are parallel and held between the parallel bases**of the trapezoid is**BC and DA, then BCDE is a parallelogram, its opposite**sides**BE and CD are equal. BE=CD.3

Consider the triangle ABE. Calculate the direction of AE. AE=AD-ED. The base

**of the trapezoid**, BC and AD are known, and in the parallelogram BCDE opposite**sides**BC and ED are equal. ED=BC, then AE=AD-BC.4

Now find out the area of triangle ABE in the formula of Heron, calculating properiter. S=sqrt(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is properiter triangle ABE. p=1/2*(AB+BE+AE). To compute the area, you are aware of all necessary data: AB, BE=CD, AE=AD-BC.

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Next, write down the area of the triangle ABE in another way - it is equal to half of the work of the triangle's height BH and

**the sides**AE, to which it is held. S=1/2*BH*AE.6

Express from this formula

*the height*of the triangle that is the height**of the trapezoid**. BH=2*S/AE. Calculate it.7

*If isosceles trapezoid, the solution is to do differently. Consider the triangle ABH. It is rectangular, as one of the corners, BHA, direct.*

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Swipe from the top C

*height*CF.9

Examine the figure of the HBCF. HBCF rectangle, since two

**sides**are the height and the other two are the bases**of the trapezoid**, that is, the straight angles, and opposite**sides**are parallel. This means that BC=HF.10

Look at right triangles ABH and FCD. The angles at the elevation BHA and CFD are straight, and the angles at the lateral

**sides**x BAH and CDF are equal, since the trapezoid ABCD is isosceles, then the triangles are similar. Because of the height BH and CF are equal or the lateral**side**of the isosceles**trapezoid**, AB and CD are equal, then these triangles are equal. Hence, their**sides**AH and FD is also equal.11

Find AH. AH+FD=AD-HF. Because of the parallelogram HF=BC, and triangles, AH=FD, then AH=(AD-BC)*1/2.

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Further, from the right triangle ABH Pythagoras to calculate

*the height*BH. The square of the hypotenuse AB is equal to the sum of the squares of the sides AH and BH. BH=sqrt(AB*AB AH*AH).# Advice 2: How to find the height of a triangle if the coordinates of the points

The height of the triangle is called a straight line connecting the top of the figure with the opposite side. This cut must be perpendicular to the side, so each vertex can hold only one

**height**. Because the peaks in this figure three, heights it the same. If a triangle specified by coordinates of its vertices, the calculation of the length of each of the heights can be produced, for example, using the formula for finding the area and calculating the lengths of the sides.Instruction

1

Assume in the calculations that the area

**of a triangle**is equal to half of a work the length of either of the parties on the length of the height lowered on this side. From this definition it follows that in order to find the height you need to know the area of the shape and the length of the side.2

Start with calculation of the lengths of the sides

**of the triangle**. Label*the coordinates*of the vertices: A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃). Then the length of the side AB, you can calculate by the formula AB = √((X₁-X₂)2 + (Y₁-Y₂)2 + (Z₁-Z₂)2). For the other two sides of these formulas are as follows: BC = √((X₂-X₃)2 + (Y₂-Y₃)2 + (Z₂-Z₃)2) and AC = √((X₁-X₃)2 + (Y₁-Y₃)2 + (Z₁-Z₃)2). For example, for a**triangle**with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) the length of the side AB will be √((3-16)2 + (5-14)2 + (7-19)2) = √(-132 + (-92) + (-122)) = √(169 + 81 + 144) = √394 ≈ 19,85. The lengths of the sides BC and AC, calculated in the same way, will be equal √(152 + 122 + 62) = √405 ≈ 20,12 and √(22 + 32 + (-62)) = √49 = 7.3

Knowledge of the lengths of the three sides, obtained in the previous step, it is enough to compute the area

**of the triangle**(S) by Heron's formula: S = ¼ * √((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). For example, after substituting in this formula the values obtained from the coordinates**of the triangle**a sample from the previous step, this formula will give a value of S = ¼ *√((19,85+20,12+7) * (20,12+7-19,85) * (19,85+7-20,12) * (19,85+20,12-7)) = ¼ *√(46,97 * 7,27 * 6,73 * 32,97) ≈ ¼ *√75768,55 ≈ ¼*275,26 = 68,815.4

Based on the area

**of the triangle**calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the height for each of the parties. Since the area is equal to half the height works for the length of the side to which it is held, to find the height divide the double area for length right side: H = 2*S/a. For of the example used above the height lowered on the side AB will be 2*68,815/16,09 ≈ 8 and 55, the height to the side of the sun will have a length of 2*68,815/20,12 if 6,84, but for AC, this value will be equal to 2*68,815/7 if of 19.66.