UPSC NDA Math Model Paper 6

Given: A (0, 0), B (0, 10), C (8, 2) and D (8, 7) are the vertices of a quadilateral ABCD Here, we have to find the area of quadilateral ABCD Area of quadilateral ABCD = Area of ΔABC + Area of Δ ACD Let's find out the area of ΔABC ∵ A (0, 0), B (0, 10), C (8, 2) are the vertices of ΔABC As we know that, if $A(x_1, y_1), B(x_2, y_2)$ and $C(x_3, y_3)$ be the vertices of a Δ ABC, then area of Δ ABC $= \frac{1}{2} \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$ $\Rightarrow$ Area of $\Delta ABC = \frac{1}{2} \cdot \begin{vmatrix} 0 & 0 & 1 \\ 0 & 10 & 1 \\ 8 & 2 & 1 \end{vmatrix}$ $\Rightarrow$ Area of $\Delta ABC = 40 \text{ sq}.$ units Similarly, let's find out the area of $\Delta ACD$ $\because A(0,0), C(8,2)$ and $D(8,7)$ $\Rightarrow$ Area of $\Delta ACD = \frac{1}{2} \cdot \begin{vmatrix} 0 & 0 & 1 \\ 8 & 2 & 1 \\ 8 & 7 & 1 \end{vmatrix}$ ⇒ Area of Δ ACD = 20 sq units ⇒ Area of quadilateral ABCD = Area of ΔABC + Area of Δ ACD = [40 + 20] sq. units = 60 sq. units Hence, option C is the correct answer.