MO circles

Juro built the ABCD square with a 12 cm side. In this square, he scattered a quarter circle with a center at point B passing through point A and a semicircle l with a center at the center of the BC side and passed point B. He would still build a circle that would lie inside the square and touch the quarter circle k, semicircle l, and side AB. Find the radius of such a circle.

Correct answer:

r =  3 cm

Step-by-step explanation:

$$\begin{gathered} |SO|=|SL|+|LO|=R+6\text{cm} \\ |SB|=|BK|-|KS|=12cm-r \\ |OE|=|OB|-|BE|=6cm-r \\ |SE|²=|SO|²-|OE|²=|SB|²-|BE|² \\ (6+r)²-(6-r)²=(12-r)²-r² \\ 12r+12r=144-24r \\ 48r=144 \\ r=144/48=3\text{cm} \end{gathered}$$

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