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Instructions

1

In problems on geometry, more precisely in planimetry andtrigonometry, sometimes it is required to find the height of the parallelogram, starting from the given values ​​of the sides, angles, diagonals, etc. To find the height of the parallelogram, knowing its area and the length of the base, it is necessary to use the rule of determining the area of ​​the parallelogram. The area of ​​the parallelogram is known to be equal to the product of the height by the length of the base: S = a * h, where: S is the area of ​​the parallelogram, a is the length of the base of the parallelogram, h is the length of the height of the heighed side a (or its extension). , that the height of the parallelogram will be equal to the area divided by the length of the base: h = S / a For example, given: the area of ​​the parallelogram is 50 sq. cm, the base - 10 cm, find: the height of the parallelogram. h = 50/10 = 5 (cm ).

2

Since the height of the parallelogram, the part of the base andthe sides of the base form a rectangular triangle, then to find the height of the parallelogram, you can use some aspect ratio and the angles of the right triangles. If the side of the parallelogram d (AD) adjacent to the height h (DE) and the angle A (BAD) opposite to the height are known parallelogram, multiply the length of the adjacent side by the sine of the opposite angle: h = d * sinA, for example, if d = 10 cm, and the angle A = 30 degrees, then H = 10 * sin (30º) = 10 * 1/2 = 5 (cm ).

3

If in the conditions of the problem the length of the adjacentthe height h (DE) of the side of the parallelogram d (AD) and the length of the part of the base cut off by the height (AE), then the height of the parallelogram can be found using the Pythagoras theorem: | AE | ^ 2 + | ED | ^ 2 = | AD | ^ 2, : h = | ED | = √ (| AD | ^ 2- | AE | ^ 2), that is, the height of the parallelogram is equal to the square root of the difference in the squares of the length of the adjacent side and the part of the base cut off by the height. For example, if the length of the adjacent side is 5 cm, and the length of the cut off part of the base is 3 cm, then the length of the height will be: h = √ (5 ^ 2- 3 ^ 2) = 4 (cm).

4

If the length of the diagonal adjacent to the height is known(DB) of the parallelogram and the length of the base part (BE) cut off by the height, then the height of the parallelogram can also be found using the Pythagorean theorem: | BE | ^ 2 + | ED | ^ 2 = | BD | ^ 2, from which we determine: h = | ED | = √ (| BD | ^ 2 | BE | ^ 2), that is, the height of the parallelogram is equal to the square root of the difference in the squares of the length of the adjacent diagonal and the part of the base cut off by the height. For example, if the length of the adjacent side is 5 cm and the length of the cut-off part of the base is 4 cm, then the length of the height will be: h = √ ( 5 ^ 2-4 ^ 2) = 3 (cm).

Tip 2: How to find high altitude

The height of a polygon is called perpendicularone of the sides of the figure is a straight line segment that connects it to the vertex of the opposing corner. There are several such segments in a plane convex figure, and their lengths are not the same if at least one of the sides of the polygon has a different value. Therefore, in problems from the course of geometry, sometimes it is required to determine the length of a larger height, for example, a triangle or a parallelogram.

Instructions

1

Determine which of the heights of the polygonshould have the longest length. In the triangle, this is a segment that is lowered to the shortest side, so if in the initial conditions the dimensions of all three sides are given, then guessing is not necessary.

2

If in addition to the length of the shortest of the sidestriangle (a) in the conditions given the area (S) of the figure, the formula for calculating the greater of the heights (Hₐ) will be fairly simple. Double the area and divide the resulting value by the length of the short side - this is the required height: Hₐ = 2 * S / a.

3

Not knowing the area, but having the lengths of all sidestriangle (a, b and c), you can also find the longest of its heights, but the mathematical operations will be much larger. Start with the calculation of the auxiliary value - semiperimeter (p). To do this, add the lengths of all sides and divide the result in half: p = (a + b + c) / 2.

4

Three times multiply the half -perimeter by the difference betweenhim and each of the parties: p * (p-a) * (p-b) * (p-c). From the value obtained, extract the square root √ (p * (p-a) * (p-b) * (p-c)) and do not be surprised - you used Heron's formula to find the area of ​​the triangle. To determine the length of the greatest height, it remains to replace the expression obtained by the area in the formula from the second step: Hₐ = 2 * √ (p * (p-a) * (p-b) * (p-c)) / a.

5

The high height of the parallelogram (Hₐ) is calculated even more simply if the area of ​​this figure (S) and the length of its short side (a) are known. Divide the first into the second and get the desired result: Hₐ = S / a.

6

If the value of the angle (α) is known in any of thevertices of the parallelogram, as well as the lengths of the sides (a and b) forming this angle, it will not be very easy to find the greater of the heights. To do this, multiply the length of the long side by the sine of the known angle, and divide the result by the length of the short side: Hₐ = b * sin (α) / a.