Respuesta :
[tex]\textit{internal division of a line segment using ratios} \\\\\\ P(-10,3)\qquad Q(a,b)\qquad \qquad \stackrel{\textit{ratio from P to Q}}{2:3} \\\\\\ \cfrac{P\underline{R}}{\underline{R} Q} = \cfrac{2}{3}\implies \cfrac{P}{Q} = \cfrac{2}{3}\implies 3P=2Q\implies 3(-10,3)=2(a,b)[/tex]
[tex](\stackrel{x}{-30}~~,~~ \stackrel{y}{9})=(\stackrel{x}{2a}~~,~~ \stackrel{y}{2b}) \\\\\\ R=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-30+2a}}{2+3}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{9+2b}}{2+3} \right)}~~ = ~~\stackrel{\textit{and we know that is}}{(4~~,~~7)} \\\\[-0.35em] ~\dotfill[/tex][tex]\cfrac{-30+2a}{5}~~ = ~~4\implies -30+2a=20\implies 2a=50\implies a=\cfrac{50}{2}\implies \boxed{a=25} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{9+2b}{5}~~ = ~~7\implies 9+2b=35\implies 2b=26\implies b=\cfrac{26}{2}\implies \boxed{b=13}[/tex]
A coordinate system in geometry is a system that employs one or more integers, or coordinates, to define the position. The coordinates of the point Q are (25,13).
What are coordinates?
A coordinate system in geometry is a system that employs one or more integers, or coordinates, to define the position of points or other geometric components on a manifold such as Euclidean space.
Let the coordinates of the point Q be represented by Q(a,b).
Given the Ratio from P to Q is 2:3.
The internal division of the line segment using the ratios will be,
PR/RQ = 2/3
3P = 2Q
3(-10, 3) = 2(a, b)
Now, the coordinates of R can be rewritten as,
[tex]R = (\dfrac{-30+2a}{2+3},\ \dfrac{9+2b}{2+3}) = (4,7)[/tex]
Comparing the coordinates,
(-30+2a)/(2+3) = 4
a = 25
(9+2b)/(2+3) = 7
b = 13
Hence, the coordinates of the point Q are (25,13).
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