Question

If the midpoints of the sides $BC$, $CA$ and $AB$ of a triangle $ABC$ are respectively $(2,1)$, $(-1,-2)$ and $(3,3)$, then the equation of the side $BC$ is

• A. $x-2y=0$
• B. $5x-4y=6$
• C. $2x+3y=8$
• D. $3x-2y=6$

Hint:

If midpoints of all three sides of a triangle are given, first find the vertices using vector relations, then write the required line equation.

Answer 1

The Correct Option is B

Step 1: Let the position vectors of the vertices $A$, $B$, and $C$ be $\vec{A}, \vec{B}, \vec{C}$. Given midpoints: \[ \frac{\vec{B}+\vec{C}}{2}=(2,1)\Rightarrow \vec{B}+\vec{C}=(4,2)\quad\cdots(1) \] \[ \frac{\vec{C}+\vec{A}}{2}=(-1,-2)\Rightarrow \vec{C}+\vec{A}=(-2,-4)\quad\cdots(2) \] \[ \frac{\vec{A}+\vec{B}}{2}=(3,3)\Rightarrow \vec{A}+\vec{B}=(6,6)\quad\cdots(3) \]
Step 2: Add equations (2) and (3): \[ 2\vec{A}+\vec{B}+\vec{C}=(4,2) \] Using equation (1), $\vec{B}+\vec{C}=(4,2)$, hence: \[ 2\vec{A}+(4,2)=(4,2)\Rightarrow \vec{A}=(0,0) \]
Step 3: From equation (3): \[ \vec{B}=(6,6) \] From equation (2): \[ \vec{C}=(-2,-4) \]
Step 4: Coordinates of $B(6,6)$ and $C(-2,-4)$ are known. Slope of $BC$: \[ m=\frac{6-(-4)}{6-(-2)}=\frac{10}{8}=\frac{5}{4} \]
Step 5: Equation of line $BC$: \[ y-6=\frac{5}{4}(x-6) \] Simplifying: \[ 4y-24=5x-30 \Rightarrow 5x-4y=6 \]