Given:
The endpoints of a line segment are F(-8, 10) and G(8, -2).
Point K partitions the segment, starting at point F [tex]\dfrac{3}{4}[/tex] of the way to point G.
To find:
The coordinates of point K.
Solution:
Section formula: If a point divides a line segment in m:n, then the coordinates of that point are
[tex]Point=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)[/tex]
According to the given information,
[tex]\dfrac{FK}{FG}=\dfrac{3}{4}[/tex]
[tex]FK:FG=3:4[/tex]
Now,
[tex]FK:KG=FK:(FG-FK)[/tex]
[tex]FK:KG=3:(4-3)[/tex]
[tex]FK:KG=3:1[/tex]
It means point K divides the segment in 3:1.
Using the section formula, we get
[tex]K=\left(\dfrac{3(8)+1(-8)}{3+1},\dfrac{3(-2)+1(10)}{3+1}\right)[/tex]
[tex]K=\left(\dfrac{24-8}{4},\dfrac{-6+10}{4}\right)[/tex]
[tex]K=\left(\dfrac{16}{4},\dfrac{4}{4}\right)[/tex]
[tex]K=\left(4,1\right)[/tex]
Therefore, the coordinates of the point K are (4,1).
