$$△AEC\approx △ADE$$

$$\frac{2·a}{2}=\frac{\frac{a}{2}}{x}$$

$$\cancel{a}·x=\frac{\cancel{a}}{2}$$

$$x^2+{\left(2a\right)}^2=2^2$$

$${\left(\frac{1}{2}\right)}^2+{\left(2a\right)}^2=2^2$$

$$\frac{1}{4}+4·a^2=4 |·4$$

$$1+16·a^2=16$$

$$16·a^2=16-1$$

$$16·a^2=15$$

$$a^2=\frac{15}{16}$$

$$\frac{y}{\frac{a}{2}}=\frac{x}{2a}$$

$$y·2·\bcancel{a}=x·\frac{\bcancel{a}}{2}$$

$$y=\frac{1}{2}·x·\frac{1}{2}$$

$$y=\frac{1}{4}·x$$

$$y=\frac{1}{4}·\frac{1}{2}$$

The radius of a circle inscribed in a triangle can be calculated most easily using the area of ​​the triangle.

$$T=\frac{a·m}{2}=r·s=r\frac{a+2a+2a}{2}$$

$$\frac{\bcancel{a}·m}{\bcancel{2}}=r·\frac{5·\bcancel{a}}{\bcancel{2}}$$

$$m=5·r$$

$$r=\frac{m}{5}$$

$$m=2-y=2-\frac{1}{8}$$

$$m=\frac{16}{8}-\frac{1}{8}$$

$$r=\frac{m}{5}=\frac{15}{8}·\frac{1}{5}$$