Question

Let $A(0,-6),B(1,2),C(3,-3),D(-3,2),$ and $E(-4,-2)$ be given points, then which of the following is (are) true?

• A. Area of pentagon formed by these points is $\frac{73}{2}$
• B. Area of triangle formed by points below x-axis is 12 square units
• C. Angle subtended by $BD$ at $A$ is ${\tan }^{-1}\frac{32}{61}$
• D. All of these

Answer 1

The correct option is D All of these

Given the points

A(0,-6), B(1,2), C(3,-3),D(-3,2) and E(-4,-2)

Area of the pentagon = Area formed by a polygon =

$\left|\frac{(x_1{y}_2-y_1{x}_2)+(x_2{y}_3-y_2{x}_3)...+(x_n{y}_1-y_n{x}_1)}{2}\right|$

where $a_1$ is the coordinate of vertex 1 and $y_n$ is the y-coordinate of the nth vertex etc.

Area $\left|\frac{\left[\right(-3\times 2)-(2\times 1\left)\right]+\left[\right(1\times -3)-(3\times 2\left)\right]+\left[\right(3\times -6)-(-3\times 0\left)\right]+\left[\right(-4\times -2)-(-2\times -3\left)\right]+\left[\right(0\times -2)-(-6\times -4\left)\right]}{2}\right|=|-\frac{73}{2}|=\frac{73}{2}$

Area of triangle formed by points below $x-axis,

Area$=\frac{1}{2}bh={}^{}\times -6=\frac{24}{2}=12$ square unit

Angle subtended by BD at $A={\tan }^{-1}\left(\frac{82}{61}\right)$