Given:
A line segment AB.
A point is
of the way from A to B.
To find:
The coordinates of that point.
Solution:
Section formula: If a point divide a line segment in m:n, then
![Point=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)](https://tex.z-dn.net/?f=Point%3D%5Cleft%28%5Cdfrac%7Bmx_2%2Bnx_1%7D%7Bm%2Bn%7D%2C%5Cdfrac%7Bmy_2%2Bny_1%7D%7Bm%2Bn%7D%5Cright%29)
From the given figure, it is clear that the two end points of the line segment are A(-4,-5) and B(12,4).
Let the unknown point be P. The point is
of the way from A to B. It means,
![\dfrac{AP}{AB}=\dfrac{3}{10}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BAB%7D%3D%5Cdfrac%7B3%7D%7B10%7D)
Let AP and AB are 3x and 10x, then
![\dfrac{AP}{PB}=\dfrac{AP}{AB-AP}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BPB%7D%3D%5Cdfrac%7BAP%7D%7BAB-AP%7D)
![\dfrac{AP}{PB}=\dfrac{3x}{10x-3x}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BPB%7D%3D%5Cdfrac%7B3x%7D%7B10x-3x%7D)
![\dfrac{AP}{PB}=\dfrac{3x}{7x}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BPB%7D%3D%5Cdfrac%7B3x%7D%7B7x%7D)
![\dfrac{AP}{PB}=\dfrac{3}{7}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BPB%7D%3D%5Cdfrac%7B3%7D%7B7%7D)
It means, point P divided the line segment in 3:7.
Using section formula, we get
![P=\left(\dfrac{3(12)+7(-4)}{3+7},\dfrac{3(4)+7(-5)}{3+7}\right)](https://tex.z-dn.net/?f=P%3D%5Cleft%28%5Cdfrac%7B3%2812%29%2B7%28-4%29%7D%7B3%2B7%7D%2C%5Cdfrac%7B3%284%29%2B7%28-5%29%7D%7B3%2B7%7D%5Cright%29)
![P=\left(\dfrac{36-28}{10},\dfrac{12-35}{10}\right)](https://tex.z-dn.net/?f=P%3D%5Cleft%28%5Cdfrac%7B36-28%7D%7B10%7D%2C%5Cdfrac%7B12-35%7D%7B10%7D%5Cright%29)
![P=\left(\dfrac{8}{10},\dfrac{-23}{10}\right)](https://tex.z-dn.net/?f=P%3D%5Cleft%28%5Cdfrac%7B8%7D%7B10%7D%2C%5Cdfrac%7B-23%7D%7B10%7D%5Cright%29)
![P=\left(0.8,-2.3\right)](https://tex.z-dn.net/?f=P%3D%5Cleft%280.8%2C-2.3%5Cright%29)
Therefore, the coordinates of the required point are (0.8, -2.3).