**Finding the Area of a Quadrilateral**

To find the area of a quadrilateral formed by four given points, we can use the Shoelace Formula. This formula is based on the concept of calculating the signed area of a polygon by using the coordinates of its vertices. The Shoelace Formula is also known as the Gauss Area Formula or the Surveyor's Formula.

The Shoelace Formula states that the area of a polygon with vertices (x1, y1), (x2, y2), (x3, y3), ..., (xn, yn) can be calculated as follows:

Area = 1/2 * |(x1*y2 + x2*y3 + x3*y4 + ... + xn*y1) - (y1*x2 + y2*x3 + y3*x4 + ... + yn*x1)|

In our case, the given points are (1, 2), (2, -3), (-3, 4), and (0, 5). Let's calculate the area using the Shoelace Formula step by step:

**Step 1: Write down the coordinates in a clockwise manner**

We need to write down the coordinates of the given points in a clockwise manner. The order of the points is important for the formula to work correctly. In our case, the clockwise order is:

(1, 2), (2, -3), (0, 5), (-3, 4)

**Step 2: Apply the Shoelace Formula**

Now, we can apply the Shoelace Formula to calculate the area of the quadrilateral:

Area = 1/2 * |(1*-3 + 2*5 + 0*4 + (-3)*2) - (2*2 + (-3)*0 + 5*(-3) + 4*1)|

Simplifying the above expression, we get:

Area = 1/2 * |-3 + 10 + 0 - 6 - 4 + 0 - 15 + 4|

Area = 1/2 * |-14|

Area = 7

Therefore, the area of the quadrilateral formed by the given points (1, 2), (2, -3), (-3, 4), and (0, 5) is 7 square units.

Quadrilateral ABCD
In which
A = ( 1 , 2 )
B = ( 2 , -3 )
C = ( -3 , 4 )
D = ( 0 , 5 )
Area of triangle ABD
1 / 2 [ -3 ( 5 + 3 ) + 0 ( -3 - 4 ) + 2 ( 4 - 5 ) ]
1 / 2 [ -3 × 8 + 0 + 2 × ( - 1 ) ]
1 / 2 [ -24 - 2 ]
1 / 2 [- 26 ]
[ - 13 ] sq. unit

Area of triangle BCD
1 / 2 [ 1 ( -3 - 5 ) + 2 ( 5 - 2 ) + 0 ( 2 + 3 ) ]
1 / 2 [ 1 ( -8) +2 ( 3 ) + 0 ]
1/ 2 [ -8 + 6 ]
1 / 2 [ -2 ]
[ - 1 ] sq. unit



Area of quadrilateral = Area of triangle ABD + Area of triangle BCD
[ -13 ] + [ -1]
- 13 -1 = | -14 |
= 14 sq. unit