In a triangle ABC, AB = 6, AC = 8, and its area is 12√2 cm squared, find the third side of the triangle
| April 29, 2021 | education
In a triangle ABC, AB = 6, AC = 8, and its area is 12√2 cm squared, find the third side of the triangle if the angle A is obtuse.
Knowing the area of the triangle and the lengths of its two sides, we determine the angle between these sides.
Savs = AB * AC * SinBAC / 2.
SinВАС = 2 * SАвс / АВ * АС = 2 * 12 * √2 / 6 * 8 = √2 / 2.
Since, by condition, the angle BAC is obtuse, then ABC = arcsin (√2 / 2) = 1350.
By the cosine theorem, we define the length of the BC side.
BC ^ 2 = AC ^ 2 + AB ^ 2 – 2 * AC * AB * Cos135.
ВС ^ 2 = 64+ 36 – 2 * 8 * 6 * (-√2 / 2) = 100 + 48 * √2.
ВС = √ (100 + 48 * √2) cm.
Answer: The length of the third side is √ (100 + 48 * √2) cm.
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