Question

Find the coordinates of point b on segment ac such that ab is 1/3 of ac

Answer 1

Coordinate Geometry

Formula:

Let the point $R$ divide $PQ$ in the ratio $l:m$ internally. Let the coordinates of $P$ and $Q$ be $(x_{1},\:y_{1})$ and $(x_{2},\:y_{2})$ respectively.

Then the coordinates of $R$ are

$\quad \big(\frac{lx_{2}+mx_{1}}{l+m},\:\frac{ly_{2}+my_{2}}{l+m}\big)$

Solution:

Let the coordinates of the points $A$ and $C$ be $(x_{1},\:y_{1})$ and $(x_{2},\:y_{2})$ respectively.

The point $B$ lies on $AC$ and $AB=\frac{1}{3}AC$, i.e., $B$ divides the line segment $AC$ internally into the ratio $1:2$.

$\therefore$ the coordinates of the point $B$ are

$\quad\big(\frac{2x_{1}+x_{2}}{1+2},\:\frac{2y_{1}+y_{2}}{1+2}\big)$

$\to \big(\frac{2x_{1}+x_{2}}{1+2},\:\frac{2y_{1}+y_{2}}{1+2}\big)$.

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What are the $x-$ and $y-$ coordinates of point $E$, which partitions the directed line segment from $A$ to $B$ into a ratio of $1:2$.

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