The picture above is a semicircle where $\displaystyle \overline{AC}$ is its diameter and $\displaystyle \overline{AC}\perp \overline{BD}$. It is known $\displaystyle \overline{AB}=4\: \text{cm}$, $\displaystyle \overline{BC}=1\: \text{cm}$, and $\displaystyle \overline{BD}=x\: \text{cm}$. Find $\displaystyle x$.

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