Question

Find the equations of the altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).

Answer 1

The vertices of ∆ABC are A (1, 4), B (−3, 2) and C (−5, −3).

Slope of AB = \[\frac{2 - 4}{- 3 - 1} = \frac{1}{2}\]

Slope of BC =  \[\frac{- 3 - 2}{- 5 + 3} = \frac{5}{2}\]

Slope of CA =  \[\frac{4 + 3}{1 + 5} = \frac{7}{6}\]

Thus, we have:
Slope of CF =  \[- 2\]

Slope of AD =  \[- \frac{2}{5}\]

Slope of BE =  \[- \frac{6}{7}\]

Hence,

\[\text { Equation of CF is } : \]

\[y + 3 = - 2\left( x + 5 \right)\]

\[ \Rightarrow 2x + y + 13 = 0\]

\[\text { Equation of AD is } : \]

\[ y - 4 = - \frac{2}{5}\left( x - 1 \right) \]

\[ \Rightarrow 2x + 5y - 22 = 0\]

\[\text { Equation of BE is : } \]

\[ y - 2 = - \frac{6}{7}\left( x + 3 \right)\]

\[ \Rightarrow 6x + 7y + 4 = 0\]