The area of the given quadrilateral ABCD is 26 sq. units.
The quadrilateral is divided into two triangles.
The area of the quadrilateral is calculated by adding the area of the triangles formed in the quadrilateral.
Consider a qudrilateral ABCD.
Then,
Area of ABCD = Area of ΔABC + Area of ΔACD
Consider a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3).
Then, the area of the ΔABC is
Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
It is given that a Quadrilateral with vertices A(-5, 4), B(0, 3), C(4, -1), and D(4, -5).
The quadrilateral is divided into two triangles as ΔABC and ΔACD.
The area of the ΔABC with vertices A(-5, 4), B(0, 3), and C(4, -1) is
A1 = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
= 1/2 |-5(3 + 1) + 0(-1 - 4) + 4(4 - 3)|
= 1/2 |-5 × 4 + 0 + 4 × 1|
= 1/2 |-20 + 4|
= 1/2 |-16|
= 1/2 × 16
= 8
Thus, the area of the ΔABC is A1 = 8 sq. units.
Similarly, the area of the ΔACD with vertices A(-5, 4), C(4, -1), and D(4, -5) is
A2 = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
= 1/2 | -5(-1 + 5) + 4(-5 - 4) + 4(4 + 1)|
= 1/2 |-5 × 4 + 4 × -9 + 4 × 5|
= 1/2 |-20 - 36 + 20|
= 1/2 |-36|
= 1/2 × 36
= 18
Thus, the area of the ΔACD is A2 = 18 sq. units.
Then the area of the quadrilateral ABCD = A1 + A2
⇒ 8 + 18 = 26 sq. units
Therefore, the area of the given quadrilateral is 26 sq. units.
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