1. The problem asks to explain the meaning of the expression $PM = \sqrt{(a - c)^2 + (b - d)^2}$ and related formulas. 2. This formula represents the distance between two points $P(a,b)$ and $M(c,d)$ in the coordinate plane. 3. The distance formula is derived from the Pythagorean theorem: the horizontal distance is $|a - c|$ and the vertical distance is $|b - d|$. 4. The distance $PM$ is given by $$PM = \sqrt{(a - c)^2 + (b - d)^2}$$ which calculates the length of the segment connecting points $P$ and $M$. 5. The expression $\sqrt{(a + c)^2 + (b + d)^2}$ is not a standard distance formula but could represent the distance from the origin to the point $(a+c, b+d)$. 6. The expression $2\sqrt{(a - c)^2 + (b - d)^2}$ is simply twice the distance $PM$. 7. The point $\left(\frac{a + c}{2}, \frac{b + d}{2}\right)$ represents the midpoint between points $P(a,b)$ and $M(c,d)$. 8. The midpoint formula is $$\text{Midpoint} = \left(\frac{a + c}{2}, \frac{b + d}{2}\right)$$ which gives the point exactly halfway between $P$ and $M$. 9. In summary, $PM$ is the distance between two points, and the midpoint formula gives the coordinates of the point halfway between them.