ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are (1,2) and that of D are (−2,6), then what is the equation of BC?

Calculation:

Given vertices A(1, 2) and D(−2, 6), where AD is the altitude from A onto BC in an equilateral triangle ABC.

Compute the slope of AD:

\(m_{AD} = \frac{\,y_D - y_A\,}{\,x_D - x_A\,} = \frac{\,6 - 2\,}{\,-2 - 1\,} = \frac{4}{-3} = -\tfrac{4}{3}.\)

Because AD ⟂ BC, the slope of BC, \(m_{BC}\), satisfies

\(m_{AD} \cdot m_{BC} = -1 \;\Longrightarrow\; \Bigl(-\tfrac{4}{3}\Bigr) \,m_{BC} = -1 \;\Longrightarrow\; m_{BC} = \frac{-1}{\, -\tfrac{4}{3}\,} = \tfrac{3}{4}.\)

Equation of the line BC 

\(y - 6 = \tfrac{3}{4}\,(x + 2).\)

\(4(y - 6) = 3(x + 2)\;\Longrightarrow\;4y - 24 = 3x + 6.\)

\(4y - 24 - 3x - 6 = 0 \)

\(\;\Longrightarrow\; -3x + 4y - 30 = 0 \;\Longrightarrow\; 3x - 4y + 30 = 0.\)

Hence, the correct answer is Option 4.