Question

Show that line $$AB$$ is perpendicular to line $$BC$$, whose $$A(1, 2), B(2, 4)$$ and $$C(0, 5)$$.

Answer 1

$$\textbf{Step 1: Find slopes of }\boldsymbol{AB\ and \ BC}\textbf{ and then show that their product is -1.}$$

$$m_{AB} = \dfrac{4 -2}{2-1}$$ $$\left[\because \boldsymbol{m=\dfrac{y_2-y_1}{x_2-x_1}}\right]$$

$$\Rightarrow m_{AB} = 2$$ $$\quad \quad \text{.....eqn(i)}$$

$$m_{BC} =\dfrac{4-5}{2-0}$$ $$\left[\because \boldsymbol{m=\dfrac{y_2-y_1}{x_2-x_1}}\right]$$

$$\Rightarrow m_{BC} =\dfrac{-1}{2}$$ $$\quad \quad \text{.....eqn(ii)}$$

$$\text{Now, consider }$$$$m_{AB} \times m_{BC} = 2 \times \dfrac{-1}{2} $$ $$\quad \quad \textbf{[from eqn(i) and eqn(ii)}]$$

$$\Rightarrow m_{AB} \times m_{BC} = -1$$

$$\Rightarrow AB \perp BC$$ $$[\because \textbf{Lines are perpendicular, if }\boldsymbol{m_1\cdot m_2=-1}]$$

$$\textbf{Hence, AB is perpendicular to line BC}$$