in our lives very often have to deal with the use of geometry in practice, for example in construction.Among the most common geometric shapes, there are trapeze.And to ensure that the project was successful and beautiful, you need proper and accurate calculation of the elements for such a figure.
What is the trapeze?This convex quadrilateral which has a pair of parallel sides, called bases of the trapezoid.But there are two other aspects that connect these grounds.They are called lateral.One of the issues relating to this figure is: "How to find the height of the trapezoid?" Just need to pay attention to the height - a segment that determines the distance from one base to another.There are several ways to determine this distance depending upon known variables.
1. Known quantities of both bases, they denote b, and k, as well as the area of the trapezoid.Using the known values to find the height of the trapezoid, in this case very easily.As is known from the geometry, the area of a trapezoid is calculated as half the sum of the product base and height.This formula is easy to deduce the unknown quantity.To do this, divide the area in half the amount of grounds.As the formula would look like this:
S = ((b + k) / 2) * h, hence h = S / ((b + k) / 2) = 2 * S / (b + k)
2. known length of the middle line, which we denote by d, and the area.For those who do not know, the middle line is the distance between the midpoints of the sides.How to find the height of the trapezoid in this case?According to the property of a trapezoid, the middle line corresponds to half the sum of the bases, i.e. d = (b + k) / 2.Again we resort to the formula area.Replacing half of the reason the value of the median line, we get the following:
S = d * h
As you can see from the resulting formula is easy to deduce the height.Dividing the area of the value of the median line, we find the desired value.We write this formula:
h = S / d
3. Known length of one side of (b) and the angle formed between that party and the largest base.The answer to the question of how to find the height of the trapezoid, have in this case.Consider the trapezium ABCD, where AB and CD are the sides, with AB = b.The largest base is AD.The angle formed by AB and AD is denoted α.From point B omit the height h on the basis of AD.Now consider the triangle ABF obtained, which is rectangular.Side AB is the hypotenuse, and BF-the leg.Because of the properties of a right triangle the ratio of the value of the hypotenuse and leg corresponds to the sine of the angle, the opposing side (BF).Therefore, on the basis of the above, to calculate the height of the trapezoid multiply the value of a certain aspect and the sine of the angle α.In a formula is as follows:
h = b * sin (α)
4. Similarly, consider the case if you know the size of the side and the angle, denoted its β, formed between himself and a smaller base.In solving this problem the angle between the known sides and height is carried out 90 ° - β.From the properties of triangles - the ratio of the length of the leg and hypotenuse corresponds to the cosine of the angle between the two.This formula is easy to deduce the value of height:
h = b * cos (β-90 °)
5. How to find the height of the trapezoid, if you know only the radius of the inscribed circle?From the definition of the circle, it relates to a single point of each base.In addition, these points are on the same line with the center of the circle.From this it follows that the distance between them is the diameter and, at the same time, the height of the trapezoid.Looks:
h = 2 * r
6. Often there are problems in which you must find the height of an isosceles trapezoid.Recall that a trapezoid with equal sides is called an isosceles.How to find the height of the isosceles trapezoid?If the diagonals perpendicular height is equal to half the sum of the bases.
But what if diagonals are not perpendicular?Consider an isosceles trapezoid ABCD.According to its properties, the bases are parallel.From this it follows that the angles at the base will be equal.Draw two heights BF and CM.Based on the foregoing, it can be argued that the triangles ABF and DCM are equal, ie, AF = DM = (AD - BC) / 2 = (bk) / 2. Now, based on the conditions of the problem, defining the known variables, and then to findaltitude, taking into account all the properties of an isosceles trapezoid.
