Question

Find the coordinates of the points which trisect $AB$ given that $A(2,1, - 3)$ and $B(5, - 8,3)$.

Answer 1

We need to understand the condition given in the problem and then we have to use appropriate formulas to find the coordinates of the points which trisect $AB$. We have to use the section formula for internal division to calculate the coordinates of the points which trisect $AB$.
Formula used:
Section formula for internal division:
Coordinates of the point $P(x,y,z)$ which divides line segment joining $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ internally in the ratio $m:n$ are given by,
$P(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})$

Complete step-by-step solution:
Let us consider points $P$ and $Q$ trisect $AB$.
Let us draw the diagram using the above information.

Therefore,
$AP = PQ = BQ$
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore $P$divides segment $AB$ in the ratio $1:2$ internally.
Let us apply section formula for internal division,
Coordinates of the point $P(x,y,z)$ are given by,
$P(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})$
Let us calculate x-coordinate of $P$,
$x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$ $......[1]$
$P$Divides segment $AB$ in the ratio $1:2$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, $m = 1,n = 2,{x_1} = 2,{x_2} = 5$
Let us put above values in equation $[1]$,
$x = \dfrac{{(1)(5) + (2)(2)}}{{1 + 2}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$x = \dfrac{9}{3}$
On performing division we get,
$x = 3$
This is the x-coordinate of $P$.
Let us calculate y-coordinate of$P$,
$y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$ $......[2]$
$P$ Divides segment $AB$ in the ratio $1:2$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, $m = 1,n = 2,{y_1} = 1,{y_2} = - 8$
Let us put above values in equation $[2]$,
$y = \dfrac{{(1)( - 8) + (2)(1)}}{{1 + 2}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$y = \dfrac{{ - 6}}{3}$
On performing division we get,
$y = - 2$
This is the y-coordinate of $P$.
Let us calculate z-coordinate of $P$,
$z = \dfrac{{m{z_2} + n{z_1}}}{{m + n}}$ $......[3]$
$P$ Divides segment $AB$ in the ratio $1:2$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, $m = 1,n = 2,{z_1} = - 3,{z_2} = 3$
Let us put above values in equation$[3]$,
$z = \dfrac{{(1)(3) + (2)( - 3)}}{{1 + 2}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$z = \dfrac{{ - 3}}{3}$
On performing division we get,
$z = - 1$
This is the z-coordinate of $P$.
Therefore coordinate of point $P$ are $(3, - 2, - 1)$
Let, $Q$ divides segment $BC$in the ratio $2:1$ internally.
Let us apply section formula for internal division,
Coordinates of the point $Q(x,y,z)$ are given by,
$Q(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})$
Let us calculate x-coordinate of$P$,
$x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$ $......[1]$
$Q$ Divides segment $BC$ in the ratio $2:1$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, $m = 2,n = 1,{x_1} = 2,{x_2} = 5$
Let us put above values in equation $[1]$,
$x = \dfrac{{(2)(5) + (1)(2)}}{{2 + 1}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$x = \dfrac{{12}}{3}$
On performing division we get,
$x = 4$
This is the x-coordinate of $Q$.
Let us calculate y-coordinate of $Q$,
$y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$ $......[2]$
$Q$ Divides segment $BC$ in the ratio $2:1$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, $m = 2,n = 1,{y_1} = 1,{y_2} = - 8$
Let us put above values in equation $[2]$,
$y = \dfrac{{(2)( - 8) + (1)(1)}}{{2 + 1}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$y = \dfrac{{ - 15}}{3}$
On performing division we get,
$y = - 5$
This is the y-coordinate of $Q$.
Let us calculate z-coordinate of $Q$,
$z = \dfrac{{m{z_2} + n{z_1}}}{{m + n}}$ $......[3]$
$Q$ Divides segment $BC$ in the ratio $2:1$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, $m = 2,n = 1,{z_1} = - 3,{z_2} = 3$
Let us put above values in equation $[3]$,
$z = \dfrac{{(2)(3) + (1)( - 3)}}{{2 + 1}}$
On performing multiplication and additions in the numerator and in the denominator we get,
$z = \dfrac{3}{3}$
On performing division we get,
$z = 1$
This is the z-coordinate of $Q$.
Therefore coordinate of point $Q$ are $(4, - 5,1)$
Therefore coordinates of the points which trisect $AB$ are $(3, - 2, - 1)$ and $(4, - 5,1)$.

Note: Coordinate of $Q$ can also be calculated using midpoint formula for segment $PB$ , as from the diagram we can see that $Q$ is the midpoint of segment $PB$ and can find the coordinate of P by the midpoint formula after finding the coordinates of point Q as P behaves as the midpoint of AQ. The midpoint formula for 3-D coordinates $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2},\dfrac{z_1+z_2}{2}\right)$