[tex]\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(2,1)\qquad C(4,7)\qquad \qquad \stackrel{\textit{ratio from A to C}}{3:2} \\\\\\ \cfrac{A\underline{B}}{\underline{B} C} = \cfrac{3}{2}\implies \cfrac{A}{C} = \cfrac{3}{2}\implies 2A=3C\implies 2(2,1)=3(4,7)\\\\[-0.35em] ~\dotfill\\\\ B=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf B=\left(\cfrac{(2\cdot 2)+(3\cdot 4)}{3+2}\quad ,\quad \cfrac{(2\cdot 1)+(3\cdot 7)}{3+2}\right)\implies B=\left( \cfrac{4+12}{5}~,~\cfrac{2+21}{5} \right) \\\\\\ B=\left(\cfrac{16}{5}~~,~~\cfrac{23}{5} \right)\implies B=\left( 3\frac{1}{5}~~,~~4\frac{3}{5} \right)[/tex]
The coordinates of B is ( 16/5, 23/5)
When a point divides a line segment externally or internally in some ratio, we use the section formula to find the coordinates of that point. It is a handy tool used to find the coordinates of the point dividing the line segment in some ratio. This section formula can also be used to find the midpoint of a line segment and for the derivation of the midpoint formula as well.
When a point on a line segment divides it into two segments, the formula used to determine the coordinates of that point is known as the section formula. Let us say, we have a point P(x,y) that divides the line segment with marked points as A (x1,y1) and B(x2,y2).
P(x, y)= (mx2 + nx1 / (m +n) , m y2+ ny1 / (m+n))
Given:
A (2, 1) and C (4, 7)
and, AB : BC= 3:2
let the coordinates of b is (x. y)
So,
x= (3*4 + 2*2)/ 3+2
x= 16/5
and, y= 3*7 + 1* 2 / 5
y= 23/5
hence, the coordinates of B are ( 16/5, 23/5)
Learn more about section formula here:
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