Projection of triangle on coordinate axes?

Question

A triangle in the $xy$-plane is such that when projected onto the $x$-axis, $y$-axis and the line $y=x$ the results are line segments whose end points are $(1,0)$ and $(5,0)$, $(0,8)$ and $(0,13)$ and $(5,5)$ and $(7.5,7.5) $ respectively. If the area of triangle is $\Delta$,

I don’t know to how to find side length from projections length?

Answer 1

Considering the inequalities, which should hold for every triangle point, namely $\begin{cases}1\le x\le 5\\8\le y\le 13\\10\le x+y\le 15\end{cases}$, we have the following figure containing all the triangle:

The border of the figure consists of $6$ straight line segments, each segment should contain a triangle vertex, or otherwise the projections would be shorter.
It leaves only $2$ possibilities:

Knowing that the figure area is $5\cdot 4-\frac{1}{2}\cdot 1\cdot 1-\frac{1}{2}\cdot 3\cdot 3=15$ we compute the area of the triangle,

1. in the first case it's $15-\frac12\cdot 4\cdot 1-\frac12\cdot 2\cdot 3-\frac12\cdot 3\cdot 1=\frac{17}{2}$
2. in the second case it's $15-\frac12\cdot 1\cdot 3-\frac12\cdot 2\cdot 3-\frac12\cdot 4\cdot 1=\frac{17}{2}$