Advice 1: How to find the height of a trapezoid if you know all sides

A trapezium is called a convex quadrilateral whose two opposite parallel sides non-parallel and the other two. If all opposite sides of a quadrilateral are pairwise parallel, then it is a parallelogram.
How to find the height of a trapezoid if you know all sides
You will need
  • - all side of the trapezoid (AB, BC, CD, DA).
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 trapezoidthat is not isosceles.
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.
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.
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.
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.
Express from this formula the height of the triangle that is the height of the trapezoid. BH=2*S/AE. Calculate it.
If isosceles trapezoid, the solution is to do differently. Consider the triangle ABH. It is rectangular, as one of the corners, BHA, direct.
Swipe from the top C height CF.
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.
Look at right triangles ABH and FCD. The angles at the elevation BHA and CFD are straight, and the angles at the lateral sidesx 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.
Find AH. AH+FD=AD-HF. Because of the parallelogram HF=BC, and triangles, AH=FD, then AH=(AD-BC)*1/2.
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.
How to find the height of a triangle if the coordinates of the points
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.
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.
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 trianglea 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.
Based on the area of the trianglecalculated 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.
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