1. The problem is to understand and apply formulas in analytic geometry. 2. Analytic geometry uses coordinates and algebra to study geometric problems. 3. A fundamental formula is the distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ 4. Another key formula is the midpoint $M$ of segment $AB$: $$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$ 5. The slope $m$ of the line through $A$ and $B$ is: $$m=\frac{y_2-y_1}{x_2-x_1}$$ 6. The equation of the line with slope $m$ passing through point $A(x_1,y_1)$ is: $$y-y_1=m(x-x_1)$$ 7. These formulas allow solving many analytic geometry problems by substituting known coordinates and simplifying. 8. For example, to find the distance between $A(1,2)$ and $B(4,6)$: $$d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$$ 9. This shows how to apply the distance formula step-by-step. 10. Understanding these formulas and their derivations helps in solving geometry problems analytically.