Why is the area of a triangle 1/2 * b * h?

This question is more easily answered with the help of multiple diagrams, but:1) Draw a rectangle, and draw a line between the top left corner and the bottom right corner. This line divides the rectangle in half. If the area of the whole rectangle is b * h, you can see that the area of each of the two triangles created by the line is 1/2 * b * h.2) This is harder to see when you draw lines from the bottom left corner and bottom right corner that meet somewhere on the opposite (top) line of the rectangle. Drawing lines like that would divide the rectangle into three triangles of varying possible sizes. In this case, we can drop a perpendicular line from the apex of the triangle down to its base. Then we have two new rectangles, half of each of which is part of the triangle. But the two rectangles together make up the original rectangle. Therefore the area of the triangle is half of the whole of the original rectangle.3) What if the triangle has an obtuse angle in it? Imagine drawing a diagonal from the top left corner of the rectangle to the bottom right corner, and another line from the top left corner to somewhere on the base of the rectangle. We'll focus on the middle triangle here, the one with an obtuse angle in it. Well, we can find this one's area by using algebra. We call the length of the base of the triangle "b", and the base length of the rectangle "l". Then the base length of the smaller triangle (to the far left) is "l-x", and its area is 1/2 * h * (l-b). The total area of the middle triangle and the smaller triangle together must be 1/2 * l * h (as together they are the same as the triangle in part 1) ), so the area of the middle triangle is [1/2 * l * h] - [1/2 * h * (l-b)]. This works out to 1/2 * b * h.