Question

If the points $A(1,2),B(4,6),C(3,5)$ are the vertices of a $\Delta ABC$, find the equation of the line passing through the midpoints of $AB$ and $BC$.

Answer 1

The points $A(1,2),B(4,6),C(3,5)$ are the vertices of $\Delta ABC$.
Find: Equation of the line passing through the midpoints of $AB$ and $BC$.
Let midpoint of $AB=P$
and midpoint of $BC=Q$
Equation of line $PQ$:
By midpoint formula.
$P(x,y),A(1,2)=(x_1,y_1),B(4,6)=(x_2,y_2)$
$x,y=\frac{x_1+x_2}{2},\frac{y_2+y_1}{2}$
$=\frac{1+4}{2},\frac{2+6}{2}$
$=\frac{5}{2},\frac{8}{2}$
$=\frac{5}{2},4$
$\therefore P(\frac{5}{2},4)$
By midpoint formula:
$Q(x,y),B(4,6)=(x_1,y_1),C(3,5)=(x_2,y_2)$
$x,y=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$
$=\frac{4+3}{2},\frac{6+5}{2}$
$=\frac{7}{2},\frac{11}{2}$
$\therefore Q(\frac{7}{2},\frac{11}{2})$
$\therefore P(\frac{5}{2},4),Q(\frac{7}{2},\frac{11}{2})$
Equation of line PQ by two point formula
$\frac{x-x_1}{x_1-x_2}=\frac{y-y_1}{y_1-y_2}$
$\frac{x-\frac{5}{2}}{\frac{5}{2}-\frac{7}{2}}=\frac{y-4}{4-\frac{11}{2}}$ ..... $[P(\frac{5}{2},4),Q(\frac{7}{2},\frac{11}{2})]$
$[x-\frac{5}{2}÷\frac{5}{2}-\frac{7}{2}]=y-4÷4-\frac{11}{2}$
$[x-\frac{5}{2}÷\frac{-2}{2}]=y-4÷\frac{8-11}{2}$
$[\frac{2x-5}{2}÷-1]=y-4÷\left(\frac{-3}{2}\right)$
$[\frac{2x-5}{2}\times -1]=(y-4)\times -\left(\frac{2}{3}\right)$
$\therefore -\frac{2x-5}{2}=-\frac{2y-8}{3}$
$6x-15=4y-16$
$6x-4y+1=0$
$\therefore$ The equation of line $PQ$ is $6x-4y+1=0$
Hence, equation of the line passing though the midpoints of $AB$ and $BC$ is $6x-4y+1=0.$