Leave AD alone the height of triangle ABC from vertex A.
In this way, AD is opposite to BC
Given:
Vertices \[\mathbf{A}\text{ }\left( \mathbf{2},\text{ }\mathbf{3} \right),\text{ }\mathbf{B}\text{ }\left( \mathbf{4},\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{and}\text{ }\mathbf{C}\text{ }\left( \mathbf{1},\text{ }\mathbf{2} \right)\]
Let slant of line \[\mathbf{BC}\text{ }=\text{ }\mathbf{m1}\]
\[\mathbf{m1}\text{ }=\text{ }\left( -\text{ }\mathbf{1}\text{ }\text{ }\mathbf{2} \right)/\left( \mathbf{4}\text{ }\text{ }\mathbf{1} \right)\]
\[\mathbf{m1}\text{ }=\text{ }-\text{ }\mathbf{1}\]
Leave incline of line AD alone m2
ad is opposite to BC
\[\mathbf{m1}\text{ }\times \text{ }\mathbf{m2}\text{ }=\text{ }-\text{ }\mathbf{1}\]
\[-\text{ }\mathbf{1}\text{ }\times \text{ }\mathbf{m2}\text{ }=\text{ }-\text{ }\mathbf{1}\]
\[\mathbf{m2}\text{ }=\text{ }\mathbf{1}\]
The condition of the line going through point (2, 3) and having an incline of 1 is
\[\mathbf{y}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{1}\text{ }\times \text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{2} \right)\]
\[\mathbf{y}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}\]
\[\mathbf{y}\text{ }\text{ }\mathbf{x}\text{ }=\text{ }\mathbf{1}\]
Condition of the height from vertex \[\mathbf{A}\text{ }=\text{ }\mathbf{y}\text{ }\text{ }\mathbf{x}\text{ }=\text{ }\mathbf{1}\]
Length of AD = Length of the opposite from A (2, 3) to BC
Condition of BC is
\[\mathbf{y}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }-\text{ }\mathbf{1}\text{ }\times \text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{4} \right)\]
\[\mathbf{y}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }-\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{4}\]
\[\mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}\text{ }\ldots \text{ }\ldots \text{ }\ldots \text{ }\ldots \text{ }\left( \mathbf{1} \right)\]
Opposite distance (d) of a line \[\mathbf{Ax}\text{ }+\text{ }\mathbf{By}\text{ }+\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{0}\] from a point (x1, y1) is given by
Presently look at condition (1) to the overall condition of line i.e., \[\mathbf{Ax}\text{ }+\text{ }\mathbf{By}\text{ }+\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{0}\] , we get
Length of AD =
\[\left[ \mathbf{where},\text{ }\mathbf{A}\text{ }=\text{ }\mathbf{1},\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{C}\text{ }=\text{ }-\text{ }\mathbf{3} \right]\]
∴ The condition and the length of the elevation from vertex An are \[\mathbf{y}\text{ }\text{ }\mathbf{x}\text{ }=\text{ }\mathbf{1}\] and
\[\surd \mathbf{2}\] units separately.