We will do this by seeing how to draw a rectangle whose area is twice that of a given triangle, and then by noticing that the length and width of the rectangle are equal to the base and the height, respectively, of the triangle.
If you have a right triangle then we can draw a rectangle with twice the area of the triangle as in the following picture.
Can we do something similar starting with any triangle? If you have a triangle such as
can you draw a rectangle whose area is twice the area of the triangle? To help figure out how to do this, draw and cut out two identical copies of a triangle on poster board. Rearrange them, cutting them into pieces if necessary, to make a rectangle.
Problem 1. Describe a procedure for drawing a rectangle whose area is twice the area of a given triangle. Illustrate your procedure by drawing three examples. Vary the sizes and shapes of the triangles you use.
Problem 2. Show why the area of a triangle is one half times the base times the height. To do this, first show that your procedure does indeed produce a rectangle of area twice that of the triangle. Next, state why the length and width of the resulting rectangle is the same as the base and height of the original triangle. A good labelled drawing will help to do this. Finally, use the fact that the area of a rectangle is the length times the width to get the formula for the area of a triangle.
Problem 3. Does your procedure work for obtuse triangles? Suppose you start with the triangle below. Determine how to use your procedure to produce a rectangle with twice the triangle's area.
