Scroogey

The small triangle at the bottom right must be right and isosceles, and the angle at the bottom right must be 45°, because the sum of angles is 180°.

This means the largest triangle is isosceles, and the bottom length is 

The we continue to the left, looking at the small triangle with sides 

science

[Daily puzzle] Fun with geometry

south_korea_ln

The small triangle at the bottom right must be right and isosceles, and the angle at the bottom right must be 45°, because the sum of angles is 180°.
This means the largest triangle is isosceles, and the bottom length is $$4$$, i.e. the units are $$\frac{1}{2}$$.

We begin with $$z^2 + z^2 = 1$$ giving $$z^2 = \frac{1}{2}$$

The we continue to the left, looking at the small triangle with sides $$y$$, $$c$$, and $$1$$:

$$
y^2 +  c^2 = 1
$$
$$
y^2 + (5 - c)^2 = 20
$$
$$
b = \frac{2}{ \sqrt{20} }
$$
$$
x^2 + \frac{1}{5} = 1
$$
$$
x^2 = \frac{4}{5}
$$

Similarly we continue for $$x$$ and $$w$$ and get

$$
\frac{17}{18}+\frac{5}{4}+\frac{100}{64}+\frac{2}{1} = \frac{47}{8}
$$