This begins a series of post concerning how the area of a triangle can be computed. This post concerns the formula that students most often remember:

Why is this formula true? Consider , and form the altitude from to line . Suppose that the length of is and that the altitude has length . Then one of three things could happen:

**Case 1**. The altitude intersects at either or . Then is a right triangle, which is half of a rectangle. Since the area of a rectangle is , the area of the triangle must be .

Knowing the area of a right triangle will be important for Cases 2 and 3, as we will act like a good MIT freshman and use this previous work.

**Case 2**. The altitude intersects at a point between and . Then and are right triangles, and so

**Case 3**. The altitude intersects at a point that is not in between and . Without loss of generality, suppose that is between and . Then and are right triangles, and so

## 1 Comment