Triangles have some interesting geometric properties that other shapes do not have. To start with, we can define the area of a triangle and see how this relates to its geometric properties.
$$ a = {hw \over 2} $$
$a$ is the area of the triangle
$h$ is the height of the triangle
$w$ is the width of the triangle along its longest side
It is important to note that the width of a triangle is defined as the length of a single side, and the height of the triangle is defined as the length of a perpendicular line drawn from that side to the opposite corner (Figure 1). In the case of a right triangle, the height is also the length of a second side perpendicular to the first side, but this is not the case with other forms of triangle.
If you bisect a triangle using a line from a corner of a triangle to the midpoint of the opposite side of the triangle, you will yield two triangles, both of which have equal areas. The areas of these triangles are each equal to 1/2 of the area of the original triangle (Figure 2). This is apparent, as you have divided the width term in the triangle area equation by 1/2, thus also divided the area by 1/2.
If you draw perpendicular line from a corner of a triangle to the opposite side of the triangle (i.e. along the line used to measure the height of the triangle) you can make some interesting observations. Splitting a triangle along the perpendicular line used to define the height yields two right triangles (Figure 3). These triangles do not have equal areas to one another, but their areas do of course add up to equal the original area of the triangle.
As the right triangles follow the rules of SOHCAHTOA, we can use these to find the height of the triangle, if given the length of the sides of the triangle and the angles of their corners. To do this, we can use the rules of SOH in the equations below, finding the opposite side of the right triangle (which is equivalent to the height of the original triangle), using the length of the hypotenuse (which is the length of a side of the original triangle), and the angle of the corner of the triangle is the angle of the corner of our resulting right triangle.
$$ \sin\theta = {o\over h} $$ $$ o=\sin\theta \times h $$
$\theta$ is the angle of a corner in a right triangle
$o$ is the length of the side opposite that angle
$h$ is the length of the hypotenuse in a right triangle
Another property we can use this line for is that we can prove that the area of a triangle is as expressed in the earlier equation. By producing a rectangle with equal area to the triangle (Figure 4). By splitting our bisecting line halfway, and drawing parallel lines from these until the sides of the triangle are met. Drawing perpendicular lines from these points yields on each side of the original bisector a rectangle and two triangles. These can be rearranged to make two distinct rectangles of equal area to the original triangle.
These interesting geometric properties are part of what makes triangles such interesting shapes.