"Prove that every right triangular region is measurable because it can be obtained as the intersection of two rectangles. Prove that every triangular region is measurable and its area is one half the product of its base and altitude."
We have to make use of the axioms for area as a set function .
Well , I was able to show that a right angled triangle is measurable. As a right angled triangle can be shown to be the intersection of two rectangles. One which shares its two sides with the two non-hypotenuse sides of the right angled triangle. And the other rectangle has the hypotenuse of the triangle as its side.
Now my query is do we use these two rectangles to calculate the area of the triangle ?
I couldn't do it that way.
Or do we have to show that a rectangle is composed of two right angled triangles which are congruent. (This one seems easier. But in Apostol's Calculus , the notion of 'congruence' is given in terms of sets of points. I.e two sets of points are congruent if their points can be put in one-to-one correspondence in such a way that distances are preserved. Proving that the two triangles are congruent in this way seems to me to be too hard. I think , Apostol doesn't want us to use the congruence proofs of Euclidean Geometry , which is justified I guess.)
In the end, I couldn't use either of the above two methods , hence I used the method of exhaustion . (Apostol used this method to calculate the area under a parabola. I applied the same method by calculating the area under a straight line , i.e the hypotenuse.)
I wanted to know how to find the area using congruence. Do we have to use the notion of 'congruence' as Apostol has given it ?