Answer:
The given coordinate pairs are (7,-5), (-5, 4), (-8, 0), and (4, -9). We can use the distance formula to find the length of each side of the quadrilateral formed by these points.
The distance between (7,-5) and (-5, 4) is sqrt((7 - (-5))^2 + ((-5) - 4)^2) = sqrt(12^2 + (-9)^2) = 15.
The distance between (-5, 4) and (-8, 0) is sqrt((-5 - (-8))^2 + (4 - 0)^2) = sqrt(3^2 + 4^2) = 5.
The distance between (-8, 0) and (4, -9) is sqrt((-8 - 4)^2 + (0 - (-9))^2) = sqrt((-12)^2 + 9^2) = 15.
The distance between (4, -9) and (7,-5) is sqrt((4 - 7)^2 + ((-9) - (-5))^2) = sqrt((-3)^2 + (-4)^2) = 5.
So the perimeter of the quadrilateral is 15 + 5 + 15 + 5 = 40.
To find the area of the quadrilateral, we can divide it into two triangles by drawing a diagonal. Let’s use the diagonal between points (7,-5) and (-8,0). The length of this diagonal is sqrt((7 - (-8))^2 + ((-5) - 0)^2) = sqrt(15^2 + (-5)^2) = sqrt(225 + 25) = sqrt(250).
Now we can use Heron’s formula to find the area of each triangle. Let’s start with the triangle formed by points (7,-5), (-8,0), and (-5,4).
The semi-perimeter of this triangle is (15 + sqrt(250) + 5)/2. Let’s call this value s.
Using Heron’s formula, the area of this triangle is sqrt(s * (s - 15) * (s - sqrt(250)) * (s - 5)).
Now let’s find the area of the other triangle formed by points (7,-5), (-8,0), and (4,-9).
The semi-perimeter of this triangle is also (15 + sqrt(250) + 5)/2, which we have already called s.
Using Heron’s formula again, the area of this triangle is also sqrt(s * (s - 15) * (s - sqrt(250)) * (s - 5)).
So the total area of the quadrilateral is 2 * sqrt(s * (s - 15) * (s - sqrt(250)) * (s - 5)).