The union of path-connected sets sharing a nonempty path-connected set $K$, is path-connected?

If a metric space $$M$$ is the union of path-connected sets $$S_{\alpha}$$, all of which have the nonempty path-connected set $$K$$ in common, is $$M$$ path-connected?

I didnt have a good idea. I know that a union of connected sets sharing a point in common is connected so, I tried to generalize this for path-connected, but I couldnt.

My idea to prove this was take $$p \in M$$ and define the set

$$X = \{x \in M \mid x\text{ can be connected to }p\text{ by a path countained in }M\}$$

and show that $$X = M$$. Can someone help me?

Take $$p,q\in M$$. Choose a point $$k\in K$$. Then there is a path in $$M$$ joining $$p$$ to $$k$$ and there is a path in $$M$$ joining $$k$$ to $$q$$. So…