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Question
Find the equations of the altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).
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Solution
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,
\[\text { Equation of CF is } : \]
\[y + 3 = - 2\left( x + 5 \right)\]
\[ \Rightarrow 2x + y + 13 = 0\]
\[\text { Equation of AD is } : \]
\[ y - 4 = - \frac{2}{5}\left( x - 1 \right) \]
\[ \Rightarrow 2x + 5y - 22 = 0\]
\[\text { Equation of BE is : } \]
\[ y - 2 = - \frac{6}{7}\left( x + 3 \right)\]
\[ \Rightarrow 6x + 7y + 4 = 0\]
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