Question

If the vertices of a triangle are $(1,-3),(4,p)$ and $(-9,7)$ and its area is $15$ sq. units, find the value(s) of $p$.

Answer 1

Area of the triangle having 3 vertices

$A=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Here, $x_1=1,y_1=-3,x_2=4,y_2=p,x_3=-9,y_3=7$

$\pm 15=\frac{1}{2}\left|\begin{array}{ccc}1& -3& 1\\ 4& p& 1\\ -9& 7& 1\end{array}\right|$

$\pm 30=p-7+3(4+9)+1(28+9p)$

$\pm 30=p-7+12+27+28+9p$

$\pm 30=10p+60$

$10p=-90,-30$

$p=-9,-3$