Question

Find the point, M, that divides segment AB into a ratio of 3:1 if A is at (-4, -2) and B is at (4, -10).
A)
(8, 2)
B)
(4, -2)
C)
(-2, 4)
D)
(2, -8)

Answer 1

\(\bf \color{blue}{A(-4,-2)\qquad B(4,-10)\qquad ratio1=3\qquad ratio2=1\qquad 3:1}\\ \quad \\ \quad \\

\cfrac{AM}{BM}=\cfrac{ratio1}{ratio2}\implies ratio2\cdot AM=ratio1\cdot BM\quad \textit{dividing by M}\\ \quad \\
ratio2\cdot A=ratio1\cdot B\implies 1(-4,-2)=3(4,-10)\\ \quad \\\qquad \color{blue}{M=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}\\ \quad \\

\qquad thus\qquad \\ \quad \\
M=\left(\cfrac{(1\cdot -4)+(3\cdot 4)}{3+1}\quad ,\quad \cfrac{(1\cdot -2)+(3\cdot -10)}{3+1}\right)\)