Question

In the triangle ABC with vertices A (2, 3), B (4, −1) and C (1, 2), find the equation and the length of the altitude from the vertex A.

Answer 1

Equation of side BC:

\[y + 1 = \frac{2 + 1}{1 - 4}\left( x - 4 \right)\]
\[ \Rightarrow x + y - 3 = 0\]

The equation of the altitude that is perpendicular to 

\[x + y - 3 = 0\]  is \[x - y + \lambda = 0\] . 

Line \[x - y + \lambda = 0\]   passes through (2, 3).

\[\therefore 2 - 3 + \lambda = 0 \Rightarrow \lambda = 1\]

Thus, the equation of the altitude from the vertex A (2, 3) is \[x - y + 1 = 0\] . Let d be the length of the altitude from A (2, 3). \[d = \left| \frac{2 + 3 - 3}{\sqrt{1^2 + 1^2}} \right|\]
\[ \Rightarrow d = \sqrt{2}\] Hence, the required distance is \[\sqrt{2}\] .