Answer:
11,486.3 square units
Explanation:
You want the area of the triangle with side lengths a=174, b=138, c=188.
Area
The area of a triangle can be found from the lengths of the three sides using Heron's formula:
A = √(s(s -a)(s -b)(s -c)) . . . . . . . where s = (a+b+c)/2, the "semiperimeter"
Calculation
The calculation is shown in the second attachment.
s = (174 +138 +188)/2 = 500/2 = 250
A = √(250(250 -174)(250 -138)(250 -188)) = √(250(76)(112)(62))
A = √131936000 ≈ 11486.3
The area of the triangle is about 11486.3 square units.
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Additional comment
An effectively equivalent way to find the area is to use the Law of Cosines to find an angle, then use the trig formula for the area.
C = arccos((a²+b²-c²)/(2ab)) = arccos(13976/48024) ≈ 73.080899°
Area = 1/2(ab·sin(C)) = 1/2(24012·0.9567166) ≈ 11486.3
We say this is "effectively equivalent" not only because it gives the same area, but because using the relation between cos(C) and sin(C), you can demonstrate that this formula gives Heron's formula for the area.