why area of triangle changes when measured as components of triangles?

Question

If we measure an area of triangle directly using a formula I got 90 square unit. But if we measure by components like 2 triangles and one rectangle and take some it counts 90.5

Why 0.5 square unit difference occur?

Any Help will be appreciated

Answer 1

Because the joint point of the two segments is not on the point of $(11,5)$. As the equation of the line is $y = \frac{9}{20} x$, hence the coordinate of the joint point is $(\frac{20\times 5}{9} = \frac{100}{9}, 5)$ which is not exactly $(11,5)$.

Answer 2

The points A$(0,0)$ , P$(11,5)$ and B$(20,9)$ are not on the same straight line. It forms a very small triangle (APB). So, the half area of the rectangle is not the sum of the colored pieces.

$a=AP=\sqrt{11^2+5^2}\quad;\quad b=PB=\sqrt{11^2+4^2}\quad;\quad c=AB=\sqrt{9^2+20^2}$

The calculus of the area of triangle (APB) is easy thanks to the Heron's formula :

$\frac14\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}=0.5$

Similar fake problem with solution on page 23 in https://fr.scribd.com/doc/15493868/Pastiches-Paradoxes-Sophismes

Answer 3

This picture is basically what happens here.

The difference is that your picture tries to make the slope of the orange and the blue triangles almost the same as the big one, hence giving the illusion of them being sub-triangles.