Question

A line $y=x+2$ is drawn on the co-ordinate plane. This line is rotated by $90^{\circ }$ clockwise about the point $(0,2)$. A line $y=-2x+10$ is drawn and a triangle is formed by these three lines. If the area of the triangle is $\Delta$, then find the value $[\Delta ]$ where $[k]$ denotes the greatest integer less than or equal to $k$.

Answer 1

Answer: 21

Given, $y=x+2$ .......(1)
Equation of CD
$y-2=-1(x-0)$
$\Rightarrow y=2-x$ .......(2)
Area of triangle $ABC=\frac{1}{2}\left|\begin{array}{ccc}0& 2& 1\\ \frac{8}{3}& \frac{14}{3}& 1\\ 8& -6& 1\end{array}\right|$
$=\frac{1}{6}\left|\begin{array}{ccc}0& 2& 1\\ 8& 14& 3\\ 8& -6& 1\end{array}\right|$
$=\frac{1}{6}\left|\begin{array}{ccc}0& 2& 1\\ 0& 20& 2\\ 8& -6& 1\end{array}\right|$
$=\frac{1}{6}|8(4-20)|=\left|\frac{8\cdot 16}{6}\right|=\left|\frac{64}{3}\right|=\Delta$
$\Rightarrow \left[\frac{64}{3}\right]=21$