Height of a triangle is called the perpendicular which is carried out from a corner to the opposite party. Height optional lies in this geometrical figure. In some types of triangles the perpendicular gets on continuation of the opposite party and it appears outside the area limited to lines. New rectangular triangles which part of parameters is known to you are anyway formed. On them it is possible to calculate height.
It is required to you
- - a triangle with the set parties;
- - pencil;
- - square;
- - properties of height of a triangle;
- - Heron's theorem;
- - formulas of the area of a triangle.
Instruction
1. Construct a triangle with the set parties. Designate it as AVS. Designate the known parties by figures or letters and, b and page the Party and lies opposite to a corner And, the parties of b and with — respectively, opposite to corners In and S. Provedite of height to all parties of a triangle and designate them as h1, h2 and h3.
2. Triangle height on three parties can be found through different formulas of its area. Remember what the area of a triangle is equal to. It is calculated by multiplication of the basis on height and division of the received result into 2. At the same time, the area can be found on Heron's formula. In this case it is equal to a square root from the work of a poluperimetr and differences it with all parties. That is a*h/2= √ p * (p-a) * (p-b) * (p-c) where h is height, p – poluperimetr, and, b, c – the parties of a triangle.
3. Find poluperimetr. It is calculated by addition of the sizes of all parties. It can be expressed formula p=(a+b+c)/2. Instead of letters substitute the corresponding numerical values. Count the difference of a poluperimetr from each of its parties.
4. Find h1 height lowered on the party of a. It can be expressed by fraction in which denominator there is a size and. The numerator of this fraction represents a square root from the work of a poluperimetr and its differences with all parties of this triangle. h1= (√ p * (p-a) * (p-b) * (p-c)) / a,
5. It is possible poluperimetr not to calculate specially, and to express the area by other option of the same formula. It is equal to a quarter of a square root from the work of the sum of all parties for the sums of everyone two from them with a size of the third party subtracted from this sum. That is S=1/4 * √ (a+b+c) * (a+b-c) * (a+c-b) * (b+c-a). Further height is calculated just as in the first case.
6. Other two heights can be calculated on the same formula. But it is possible to use also that the relation of heights is among themselves connected with the relation of the relevant parties and it can be expressed by formula h1: h2=1/a: 1/b. h1 is already known to you, and the parties of an and b are set in conditions. Therefore solve a proportion, having multiplied h1 and 1/and and having divided all this on 1/b. In precisely the same way through any of already known heights it is possible to find also the third party.