# The area of the isosceles triangle is 196√3. The angle opposite the base is 120 °. Find the length of the side

May 31, 2020 | Education

| 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 = 196√3

AB2 / 4 = 196

AB ^ 2 = 784

AB = 28

Answer: 28

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