There are many ways to find the altitude of a triangle; so I will create a generalized solution path for you to find the altitude. Firstly we should know that there are many different kinds of triangles; Equilateral Triangles, Isosceles Triangles, Right Triangles, and some triangles that have no pattern. Some may think it is impossible to create a fool proof method that will work for every kind of triangle; but one thing we can do is turn every triangle into a Right Triangle.
To do this we will split the base of one side in half and draw a vertical line into the vertex of the angle opposite to the base. Then from this right triangle we can have two methods depending on what information we are given. If we are given the area or sides of the triangle we can easily use a Euclidian Geometry approach using a basic equation like the area function or Pythagorean Theorem. If we are given an angle instead of a side we would most likely have to take a more trigonometric approach and use some trig function line sine, cosine, or tangent.
Given Side and Area:
We know the area function so we simple plug in the area and base into the rearranged function to solve for altitude.
Altitude = 2*Area/Base
Given Two Sides:
If we know two sides we can use Pythagorean Theorem to solve for the altitude. Usually you will have to split the base into different parts and solve from those parts; but sometimes the problem will be an easy triangle like an Isosceles, or Equilateral where you can just split one side in half and then.
Altitude = (Hypotenuse^2 – Base^2)^(1/2)
Given Side and Angle:
Depending on what sides and angles of the triangle of the triangle you are given you will have to use different equations. But they will all look similar in the fact that you will be multiplying the hypotenuse of one side of the triangle with the trigonometric function of some angle on that same side.
h = Sin( Angle C)*a
h= Cos(Angle B)*a
When working out these problems remember that they are like puzzles and you can use what you are given to find other information. For example you can find the third angle if you are given the two others (Angle C = 180 – (Angle A + Angle B)). When you are given these problems you are usually given two pieces of information and almost always from those two things you will be able to calculate everything about that given triangle using different algebraic methods and some creative thinking.