Triangle ABC v2

Area of the triangle is 12 cm square. Angle ACB = 30º , AC = (x + 2) cm, BC = x cm. Calculate the value of x.

Result

x =  -8 cm

Solution:

S=12 sin30=h/x h=xsin(30) S=1/2(x+2)h S=1/2(x+2)x.sin(30) 12=1/2 0.5 x(x+2)  12=1/2 0.5 x (x+2) 0.25x20.5x+12=0 0.25x2+0.5x12=0  a=0.25;b=0.5;c=12 D=b24ac=0.5240.25(12)=12.25 D>0  x1,2=b±D2a=0.5±12.250.5=1±49 x1,2=1±7 x1=6 x2=8   Factored form of the equation:  0.25(x6)(x+8)=0 x>0,x>2 x=x2=(8)=8 cmS=12 \ \\ \sin 30^\circ =h/x \ \\ h=x \sin(30^\circ ) \ \\ S=1/2 (x+2)h \ \\ S=1/2 (x+2)x. \sin(30^\circ ) \ \\ 12=1/2 \cdot \ 0.5 \ x(x+2) \ \\ \ \\ 12=1/2 \cdot \ 0.5 \ x \cdot \ (x+2) \ \\ -0.25x^2 -0.5x +12=0 \ \\ 0.25x^2 +0.5x -12=0 \ \\ \ \\ a=0.25; b=0.5; c=-12 \ \\ D=b^2 - 4ac=0.5^2 - 4\cdot 0.25 \cdot (-12)=12.25 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ -0.5 \pm \sqrt{ 12.25 } }{ 0.5 }=-1 \pm \sqrt{ 49 } \ \\ x_{1,2}=-1 \pm 7 \ \\ x_{1}=6 \ \\ x_{2}=-8 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 0.25 (x -6) (x +8)=0 \ \\ x>0 , x>2 \ \\ x=x_{2}=(-8)=-8 \ \text{cm}

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