You will need

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

Instruction

1

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.

5

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.*

8

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.

12

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).