The area of the isosceles triangle is 1600√3. The angle opposite the base is 120 °. Find the length of the side

We will mark the key points as shown in the figure and draw the height BD.
The height of BD is also a median and a bisector (according to the third property of an isosceles triangle).
Triangle area ABC SABC = (1/2) AC * BD
Since BD is the median, AC = 2AD
Then:
SABC = (1/2) 2AD * BD = AD * BD
Since BD is also a bisector, ∠ABD = ∠ABC / 2 = 60 °
AD = AB * sin (∠ABD) = AB * sin60 °
BD = AB * cos (∠ABD) = AB * cos60 °
Then:
SABC = AB * sin60 ° * AB * cos60 ° = AB2 (√3 / 2) * (1/2) = AB2√3 / 4 = 1600√3
AB ^ 2/4 = 1600
AB ^ 2 = 1600 * 4
AB = 40 * 2 = 80
Answer: 80

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