Answer:
[tex]P\left(\frac{5}{2},\frac{5}{2}\right)[/tex] is the point that partitions line segment ((1,1),(6,6)) into a 3:7 ratio.
Step-by-step explanation:
Let AB is the line segment with A having the coordinates (1, 1) and B having the coordinates (6, 6)
Let us consider the point P(x, y) partitions the line segment AB into a ratio 3:7
Considering the formula to find the coordinates of P(x, y)
From the data,
[tex]m_1=3,\:\:m_2=7,\:\:x_1=1,\:\:x_2=6,\:\:y_1=1,\:\:y_2=6[/tex]
so
[tex]x=\frac{m_1x_2+m_2x_1}{m_1+m_2}[/tex]
[tex]x=\frac{\left(3\right)\left(6\right)+\left(7\right)\left(1\right)}{\left(3+7\right)}[/tex]
[tex]x=\frac{25}{3+7}[/tex]
[tex]x=\frac{25}{10}[/tex]
[tex]x=\frac{5}{2}[/tex]
also
[tex]y=\frac{m_1y_2+m_2y_1}{m_1+m_2}[/tex]
[tex]y=\frac{\left(3\right)\left(6\right)+\left(7\right)\left(1\right)}{\left(3+7\right)}[/tex]
[tex]\:y=\frac{25}{3+7}[/tex]
[tex]y=\frac{25}{10}[/tex]
[tex]y=\frac{5}{2}[/tex]
Therefore, [tex]P\left(\frac{5}{2},\frac{5}{2}\right)[/tex] is the point that partitions line segment ((1,1),(6,6)) into a 3:7 ratio.
