1. **Problem Statement:** Find the ratio in which a point divides a line segment and determine the midpoint of the segment. 2. **Formula for Ratio:** If a point $P(x,y)$ divides the line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$, then $$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$$ 3. **Midpoint Formula:** The midpoint $M$ of the segment $AB$ is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ 4. **Explanation:** To find the ratio, set the coordinates of $P$ equal to the formula above and solve for $m:n$. For the midpoint, simply average the $x$ and $y$ coordinates of $A$ and $B$. 5. **Example:** Suppose $A(2,3)$, $B(8,7)$, and $P(5,5)$ lies on $AB$. Find the ratio $m:n$. 6. Using the formula for $x$ coordinate: $$5 = \frac{m \cdot 8 + n \cdot 2}{m+n}$$ Multiply both sides by $m+n$: $$5(m+n) = 8m + 2n$$ $$5m + 5n = 8m + 2n$$ Rearranged: $$8m - 5m = 5n - 2n$$ $$3m = 3n \implies m = n$$ 7. Using the formula for $y$ coordinate: $$5 = \frac{m \cdot 7 + n \cdot 3}{m+n}$$ Multiply both sides by $m+n$: $$5(m+n) = 7m + 3n$$ $$5m + 5n = 7m + 3n$$ Rearranged: $$7m - 5m = 5n - 3n$$ $$2m = 2n \implies m = n$$ 8. Both coordinates confirm $m:n = 1:1$, so $P$ is the midpoint. 9. Calculate midpoint: $$M = \left(\frac{2+8}{2}, \frac{3+7}{2}\right) = (5,5)$$ **Final answer:** The point $P$ divides the segment $AB$ in the ratio $1:1$, and $P$ is the midpoint of $AB$ at $(5,5)$.