The vertices of a convex pentagon (in order) are P(0,0),  Q(6,0),  R(8,3),  S(4,7),  T(0,4). Find the area of the pentagon.

The correct option is 39 sq. units.

To find the area of the convex pentagon with vertices taken in order as $P(0,0)$, $Q(6,0)$, $R(8,3)$, $S(4,7)$, and $T(0,4)$, we can use the Polygon Area Formula (also known as the Shoelace Formula).

The formula for the area of a polygon with vertices $(x_1,y_1)$, $(x_2,y_2)$, ..., $(x_n,y_{\mathrm{n}})$ in counterclockwise order is given by:

$\text{Area}=\frac{1}{2}\mid (x_1{y}_2+x_2{y}_3+x_3{y}_4+x_4{y}_5+x_5{y}_1)-(y_1{x}_2+y_2{x}_3+y_3{x}_4+y_4{x}_5+y_5{x}_1)\mid$

Let us write down the coordinates in sequence:
$(x_1,y_1)=(0,0)$
$(x_2,y_2)=(6,0)$
$(x_3,y_3)=(8,3)$
$(x_4,y_4)=(4,7)$
$(x_5,y_5)=(0,4)$

Step 1: Calculate the sum of products of adjacent coordinates in the main diagonal direction:
$x_1{y}_2+x_2{y}_3+x_3{y}_4+x_4{y}_5+x_5{y}_1$
$=(0\times 0)+(6\times 3)+(8\times 7)+(4\times 4)+(0\times 0)$
$=0+18+56+16+0=90\}$

Step 2: Calculate the sum of products of adjacent coordinates in the opposite diagonal direction:
$y_1{x}_2+y_2{x}_3+y_3{x}_4+y_4{x}_5+y_5{x}_1$
$=(0\times 6)+(0\times 8)+(3\times 4)+(7\times 0)+(4\times 0)\}$
$=0+0+12+0+0=12\}$

Step 3: Compute the area:
$\text{Area}=\frac{1}{2}\mid 90-12\mid =\frac{1}{2}\mid 78\mid =39$

Thus, the area of the pentagon is 39 sq. units.