40 Points ♥ Trigonometry

The area of an isosceles triangle is 100cm².
Calculate the perimeter of the triangle given that one of the angles is π/6 rad.

Case 1: 50.35 cm

Case 2: 56.715 cm

Step-by-step explanation:

AB = AC;

½ X AH x BC = 100 cm2

=> AH x BH = 100 cm2  

=> AH = 100/BH

Case 1:  (BAC)=  /6

S = ½ x AB x AC x sin (BAC) = 100 cm2

AB^2 x sin /6 = 200  

=> AB^2 x ½ = 200  

=> AB = 20 cm = AC

BC^2 = AB^2 + AC^2 – 2 x AB x AC x cos (BAC) = 202 + 202 – 2 x 20 x 20 x /2 = (800 - 400) cm

=> BC = 10.35 cm

Perimeter: AB + AC + BC = 20 + 20 + 10.35 = 50.35 cm

Case 2:  (ABC) =  /6 => (BAC) =  2/3

S = ½ x AB x AC x sin (BAC) = 100 cm2

AB^2 x sin 2/3 = 200  

=> AB^2 x /2 = 200  

=> AB^2 = 400/

=> AB = AC = 15.197 cm

BC^2 = AB^2 + AC^2 – 2 x AB x AC x cos (BAC) = 400/  + 400/ – 2 x 400/  x (-1/2) = 692.82 cm

=> BC = 26.321

Perimeter: AB + AC + BC = 15.197 + 15.197 + 26.321 = 56.715 cm

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