Height of a polygon call a straight line piece perpendicular to one of the parties of a figure which connects it to top of an opposite corner. Such pieces in a flat convex figure there is several, and their length are not identical if at least one of the parties of a polygon has the size, other than others. Therefore in tasks from a geometry course sometimes it is required to determine length of bigger height, for example, of a triangle or a parallelogram.
Instruction
1. Determine which of heights of a polygon has to have the greatest length. In a triangle it is the piece lowered on the shortest party, therefore if in initial conditions the sizes of all three parties are given, then it is not necessary to guess.
2. If except length of the shortest of the parties of a triangle (a) the area (S) of a figure, a formula of calculation bigger of heights is specified in conditions (H ₐ) it will be rather simple. Double the area and divide the received value into length of the short party - it is and there will be required height: H ₐ = 2*S/a.
3. Without knowing the area, but having lengths of all parties of a triangle (a, b and c), it is possible to find the longest of its heights too, however mathematical operations will be much more. Begin with calculation of auxiliary size - a poluperimetr (p). For this purpose put lengths of all parties and halve result: р = (a+b+c)/2.
4. Increase poluperimetr by the difference between it and each of the parties three times: r * (r-a) * (r-b) * (r-c). Take a square root from the received value √ (r * (r-a) * (r-b) * (r-c)) and be not surprised - you used Heron's formula for finding of the area of a triangle. For determination of length of the greatest height it was necessary to replace with the received expression the area in a formula from the second step: H ₐ = 2 * √ (r * (r-a) * (r-b) * (r-c)) / a.
5. Big height of a parallelogram (H ₐ) is calculated even more simply if the area of this figure (S) and length of its short party (a) is known. Divide the first into the second and receive the necessary result: H ₐ = S/a.
6. If corner size (α) in any of parallelogram tops and also length of the parties is known (an and b), forming this corner, it will be not really simple to find big of heights too. For this purpose increase the size of the long party by a sine of the known corner, and divide result into length of the short party: H ₐ = b*sin(α)/a.