Question

$AD,BE$ and $CF$ are the altitudes of a triangle $ABC$ whose vertex $A$ is the point $(-4,5)$. The coordinates of the points $D,E,F$ are respectively $(\frac{16}{5},-\frac{23}{5}),(4,1)$ and $(-1,-4)$. Find the equation of the sides of the triangle and the coorindates of the other two vertices of the triangle.

Answer 1

Slope of $AD$ is $-\frac{4}{3}$ so that slope of $BC$ is $\frac{3}{4}$ and it passes through $D$ so that its equation is
$3x-4y-28=\mathrm{0....}\left(1\right)$ by $y-y_1=m(x-x_1)$
Equation of $AB\left(AF\right)$ by two point formula is
$3x+y+7=\mathrm{0.......}\left(2\right)$
and of $AC$ is $x+2y-6=\mathrm{0......}\left(3\right)$
Solving (1), (2) and (2), (3) the coordinates of vertices $B$ and $C$ are found to be $(0,-7)$ and $(8,-1)$ respectively.