Question

Demonstrate/explain how you can use the formula that partitions any segment into a given ratio to derive the specific formula for a midpoint. (Hint: Think of the ratio that is formed by a midpoint.)

Answer 1

let's say we have two points A(a , b) and B(c , d) partitioned by a point C, since we're dealing with the MidPoint, C is partitioning AB into two equal halves, so in the ratio of 1 : 1.

`\textit{internal division of a line segment using ratios} \\\\\\ A(a,b)\qquad B(c,d)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:1} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{1}\implies \cfrac{A}{B} = \cfrac{1}{1}\implies 1A=1B \\\\\\ 1(a,b)=1(c,d) \implies (a~~,~~ b)=(c~~,~~ d) \\\\\\ C=\left( \cfrac{a+c}{1+1}~~,~~\cfrac{b+d}{1+1} \right)\implies \stackrel{ \textit{MidPoint Formula} }{C=\left( \cfrac{a+c}{2}~~,~~\cfrac{b+d}{2} \right)}`