Given:
Line segment AB has one endpoint at A(0,0).
(5, 3) is 1/3 of the way from A to B.
To find:
The coordinates of point B.
Solution:
Let the coordinates of point B are (a,b).
Suppose point P(5, 3) is 1/3 of the way from A to B.
![\dfrac{AP}{AB}=\dfrac{1}{3}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BAB%7D%3D%5Cdfrac%7B1%7D%7B3%7D)
![\dfrac{AP}{PB}=\dfrac{AP}{AB-AP}=\dfrac{1}{3-1}=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7BAP%7D%7BPB%7D%3D%5Cdfrac%7BAP%7D%7BAB-AP%7D%3D%5Cdfrac%7B1%7D%7B3-1%7D%3D%5Cdfrac%7B1%7D%7B2%7D)
It means, point P(5, 3) divides the segment AB in 1:2.
Section formula:
If a point divides 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)
Point P(5, 3) divides the segment AB in 1:2. Using section formula, we get
![P=\left(\dfrac{1(a)+2(0)}{1+2},\dfrac{1(b)+2(0)}{1+2}\right)](https://tex.z-dn.net/?f=P%3D%5Cleft%28%5Cdfrac%7B1%28a%29%2B2%280%29%7D%7B1%2B2%7D%2C%5Cdfrac%7B1%28b%29%2B2%280%29%7D%7B1%2B2%7D%5Cright%29)
![(5, 3)=\left(\dfrac{a}{3},\dfrac{b}{3}\right)](https://tex.z-dn.net/?f=%285%2C%203%29%3D%5Cleft%28%5Cdfrac%7Ba%7D%7B3%7D%2C%5Cdfrac%7Bb%7D%7B3%7D%5Cright%29)
On comparing both sides, we get
![\dfrac{a}{3}=5](https://tex.z-dn.net/?f=%5Cdfrac%7Ba%7D%7B3%7D%3D5)
![a=15](https://tex.z-dn.net/?f=a%3D15)
![\dfrac{b}{3}=3](https://tex.z-dn.net/?f=%5Cdfrac%7Bb%7D%7B3%7D%3D3)
![b=9](https://tex.z-dn.net/?f=b%3D9)
Therefore, the coordinates of point B are (15,9).