Trapeze is called the convex quadrangle at which two opposite sides are parallel and two others are nonparallel. If all opposite sides of a quadrangle are in pairs parallel, then it is a parallelogram.
It is required to you
- - all parties of a trapeze (AB, BC, CD, DA).
Instruction
1. The nonparallel parties of a trapeze are called sides, and parallel - the bases. The line between the bases, perpendicular to them - trapeze height. If sides of a trapeze are equal, then it is called isosceles. At first we will consider the decision for a trapeze which is not isosceles.
2. Carry out BE piece from B point to the lower basis of AD parallel to CD trapeze side. As BE and CD are parallel and carried out between the parallel bases of a trapeze of BC and DA, BCDE is a parallelogram, and its opposite sides of BE and CD are equal. BE=CD.
3. Consider ABE triangle. Calculate the party of AE. AE=AD-ED. The bases of a trapeze of BC and AD are known, and the opposite parties of ED and BC are equal in BCDE parallelogram. ED=BC, so AE=AD-BC.
4. Now recognize the area of a triangle of ABE by Heron's formula, having calculated poluperimetr. S= root (p * (PAB) * (p-BE) * (p-AE)). In this formula p - poluperimetr ABE triangle. p=1/2 * (AB+BE+AE). For calculation of the area all necessary data are known to you: AB, BE=CD, AE=AD-BC.
5. Further write down the area of a triangle of ABE in a different way - it is equal to a half of the work of height of a triangle of BH and the party of AE to which it is carried out. S=1/2*BH*AE.
6. Express the triangle height which is also trapeze height from this formula. BH=2*S/AE. Calculate it.
7. If a trapeze isosceles, the decision can be executed in a different way. Consider ABH triangle. It is rectangular as one of corners, BHA, straight line.
8. Carry out the C height of CF from top.
9. Study HBCF figure. HBCF a rectangle as two of its parties are heights, and others two are the trapeze bases, that is right angles, and the opposite parties are parallel. It means that BC=HF.
10. Look at rectangular triangles of ABH and FCD. Corners with heights of BHA and CFD straight lines, and corners at side storony BAH and CDF are equal as ABCD trapeze isosceles, so triangles are similar. As heights of BH and CF are equal or sides of an isosceles trapeze of AB and CD are equal, and similar triangles are equal. Means, their parties of AH and FD are equal too.
11. Find AH. AH+FD=AD-HF. As from HF=BC parallelogram, and from triangles of AH=FD, (AH=AD-BC) *1/2.
12. Further from a rectangular triangle of ABH on Pythagorean theorem calculate BH height. The square of a hypotenuse of AB is equal to the sum of squares of legs of AH and BH. BH= root (AB*AB-AH*AH).