Height of a triangle is called the perpendicular which is carried out from triangle top to the straight line containing the opposite side. Length of height can be determined in two ways. The first - from the area of a triangle. The second - considering height as a leg of a rectangular triangle.
It is required to you
- - handle;
- - note paper;
- - calculator.
Instruction
1. The first way to find height – through the area of a triangle. The area of a triangle is calculated on a formula: S = 1/2 ah where (a) is the party of a triangle, h are height constructed to the party (a). From this expression find height: h = 2S/a.
2. If in a condition lengths of three parties of a triangle are given, find the area on Heron's formula: S = (p * (p-a) * (p-b) * (p-c)) ^1/2, where p – poluperimetr a triangle; and, b, with – its parties. Knowing the area, you can determine height length to any party.
3. For example, in a task the perimeter of a triangle in which the circle with the known radius is entered is specified. Calculate the area from expression: S = r*p where r is the radius of an inscribed circle; p – poluperimetr. From the area calculate height to the party which length is known to you.
4. The area of a triangle can also be determined on a formula: S = 1/2ab*sina where and, b are the parties of a triangle; sina is a sine of the angle between them.
5. One more case – all corners of a triangle and one party are known. Use the theorem of sine: a/sina = b/sinb = with / sinc = 2R where a, b, c are the parties of a triangle; sina, sinb, sinc are sine of corners, opposite to these parties; R – radius of a circle which can be described around a triangle. Find the party of b from a ratio: a/sina = b/sinb. Then calculate the area similar to a step 4.
6. The second way to calculate height is to apply trigonometrical dependences to a rectangular triangle. Height in an acute triangle divides it into two rectangular. If the party, opposite is known to the basis (a), and a corner between them, apply expression: h = b*sina. In a formula changes a little: h = b*sin(180-a) or h = - c*sina.
7. If you were given opposite to height a corner and length of a piece of AH which height cuts from the basis, use dependence: BH = (AH) of *tga.
8. Also, knowing lengths of a piece of AH and the party of AV, find VN height from Pythagorean theorem: BH = (AB^2 – BC^2) ^1/2.