Respuesta :
so, we know the segment MP gets partitioned by the point A to MA with a ratio of 2 and AP with a ratio of 1, on a 2:1 ratio from M to P, therefore then,
[tex]\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ M(1,4)\qquad P(16,14)\qquad \qquad 2:1 \\\\\\ \cfrac{MA}{AP} = \cfrac{2}{1}\implies \cfrac{M}{P} = \cfrac{2}{1}\implies 1M=2P\implies 1(1,4)=2(16,14)\\\\ -------------------------------\\\\ { A=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}[/tex]
[tex]\bf -------------------------------\\\\ A=\left(\cfrac{(1\cdot 1)+(2\cdot 16)}{2+1}\quad ,\quad \cfrac{(1\cdot 4)+(2\cdot 14)}{2+1}\right) \\\\\\ A=\left( \cfrac{1+32}{3}~~,~~\cfrac{4+28}{3} \right)\implies A=\left( \cfrac{33}{3}~~,~~\cfrac{32}{3} \right) \\\\\\ A=\left(11~~,10\frac{2}{3} \right)[/tex]
[tex]\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ M(1,4)\qquad P(16,14)\qquad \qquad 2:1 \\\\\\ \cfrac{MA}{AP} = \cfrac{2}{1}\implies \cfrac{M}{P} = \cfrac{2}{1}\implies 1M=2P\implies 1(1,4)=2(16,14)\\\\ -------------------------------\\\\ { A=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}[/tex]
[tex]\bf -------------------------------\\\\ A=\left(\cfrac{(1\cdot 1)+(2\cdot 16)}{2+1}\quad ,\quad \cfrac{(1\cdot 4)+(2\cdot 14)}{2+1}\right) \\\\\\ A=\left( \cfrac{1+32}{3}~~,~~\cfrac{4+28}{3} \right)\implies A=\left( \cfrac{33}{3}~~,~~\cfrac{32}{3} \right) \\\\\\ A=\left(11~~,10\frac{2}{3} \right)[/tex]
The coordinate of point A that divide a line segment of MP in the ratio of MA: AP = 2: 1 would be (11, 32/3).
What is the section formula?
The section formula is used to find the coordinates of a point when the point divides a line segment externally or internally in some ratio.
Let's consider a point P(x, y) that divides a line with endpoints as A and B.
[tex]\rm P(x ,y) = ( \frac{mx_{2}+ nx_{1}}{m+n} ,\frac{my_{2}+ ny_{1}}{m+n})[/tex]
Here, m and n are the ratios of division
It is given that the endpoints of MP are M(1,4) and P(16,14). If A partitions MP in a ratio of MA: AP = 2: 1
[tex]\rm A(x ,y) = ( \frac{mx_{2}+ nx_{1}}{m+n} ,\frac{my_{2}+ ny_{1}}{m+n})[/tex]
A(x, y) = 2(16) + 1(1)/2+1 , 2(14) + 1(4)/2+1
A(x, y) = 32+1/3 , 28 +4 /3
A(x, y) = 11, 32/3
Thus, the coordinates of point A(x, y) = (11, 32/3)
Learn more about the section formula;
https://brainly.com/question/11888968
