Find the area of the triangle with vertices (−5,−5), (4,−4), and (−7,3).

Find the area of the triangle with vertices (−5,−5), (4,−4), and (−7,3).

Given sides a,b,c: s = (a + b + c)/2.
Then the area is √(s(s - a)(s - b)(s - c)).
This is Heron's formula.

a = d((-5,-5),(4,-4)) = √((-5 - 4)^2 + (-5 + 4)^2) = √(81 + 1) = √82
b = d((-5,-5),(-7,3)) = √((-5 + 7)^2 + (-5 - 3)^2) = √(4 + 64) = √68
c = d((4,-4),(-7,3)) = √((4 + 7)^2 + (-4 - 3)^2) = √(121 + 49) = √170
s = (√82 + √68 + √170)/2
s - a = (√68 + √170 - √82)/2
s - b = (√82 + √170 - √68)/2
s - c = (√68 + √82 - √170)/2
√(s(s - a)(s - b)(s - c)) = √1369 = 37

Find the distance between each point:

A^2 = (-5 - 4)^2 + (-5 - (-4))^2
A^2 = (-9)^2 + (-5 + 4)^2
A^2 = 81 + 1
A^2 = 82
A = sqrt(82)

B^2 = (-5 - (-7))^2 + (-5 - 3)^2
B^2 = (-5 + 7)^2 + (-8)^2
B^2 = 4 + 64
B^2 = 68
B = sqrt(68)
B = 2 * sqrt(17)

C^2 = (4 - (-7))^2 + (-4 - 3)^2
C^2 = (4 + 7)^2 + (-7)^2
C^2 = 11^2 + (-7)^2
C^2 = 121 + 49
C^2 = 170
C = sqrt(170)


s = (A + B + C) / 2
s = (sqrt(82) + 2 * sqrt(17) + sqrt(170)) / 2

A = sqrt(s * (s - A) * (s - B) * (s - C))

s - A => (-sqrt(82) + 2 * sqrt(17) + sqrt(170)) / 2
s - B => (sqrt(82) - 2 * sqrt(17) + sqrt(170)) / 2
s - C => (sqrt(82 + 2 * sqrt(17) - sqrt(170)) / 2

(1/16) * (sqrt(82) + 2 * sqrt(17) + sqrt(170)) * (-sqrt(82) + 2 * sqrt(17) + sqrt(170)) * (sqrt(82) - 2 * sqrt(17) + sqrt(170)) * (sqrt(82) + 2 * sqrt(17) - sqrt(170))

Let's take this piece by piece:

(2 * sqrt(17) + sqrt(170) + sqrt(82)) * (2 * sqrt(17) + sqrt(170) - sqrt(82)) =>
(4 * 17 + 2 * sqrt(17) * sqrt(170) - 2 * sqrt(17) * sqrt(82) + 2 * sqrt(17) * sqrt(170) + 170 - sqrt(82) * sqrt(170) + 2 * sqrt(17) * sqrt(82) + sqrt(170) * sqrt(82) + 82) =>
(68 + 2 * 17 * sqrt(10) - 2 * sqrt(17 * 82) + 2 * 17 * sqrt(10) + 170 - sqrt(82 * 170) + 2 * sqrt(17 * 82) + sqrt(82 * 170) + 82) =>
(68 + 34 * sqrt(10) - 2 * sqrt(17 * 82) + 34 * sqrt(10) + 170 - sqrt(82 * 170) + 2 * sqrt(17 * 82) + sqrt(82 * 170) + 82) =>
(68 + 170 + 82 + 68 * sqrt(10)) =>
(320 + 68 * sqrt(10)) =>
4 * (80 + 17 * sqrt(10))


(sqrt(82) - 2 * sqrt(17) + sqrt(170)) * (sqrt(82) + 2 * sqrt(17) - sqrt(170))
82 + 2 * sqrt(17 * 82) - sqrt(82 * 170) - 2 * sqrt(17 * 82) - 4 * 17 + 2 * 17 * sqrt(10) + sqrt(82 * 170) + 2 * 17 * sqrt(10) - 170 =>
82 - 68 + 34 * sqrt(10) + 34 * sqrt(10) - 170) =>
68 * sqrt(10) - 156 =>
4 * (17 * sqrt(10) - 39)

(1/16) * 4 * 4 * (17 * sqrt(10) + 80) * (17 * sqrt(10) - 39)
(289 * 10 + 80 * 17 * sqrt(10) - 39 * 17 * sqrt(10) - 3120)
(2890 + 17 * sqrt(10) * (80 - 39) - 3120)
(41 * 17 * sqrt(10) - 230)
1974.1075291373603944032288004696

Take the square root of it:

44.430929870275733341332754461145