A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 27, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 27, what is the area of the triangle?

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Answer 1 · 2018-07-15T19:02:46

$A = \frac{1}{4} \cdot 27^{2} \cdot \sqrt{2} \cdot (\sqrt{3} + 1)$ $A=1/4*27^2*sqrt(2)*(sqrt(3)+1)$

Explanation:

At first we will compute the third angle:
$π - \frac{3}{4} \cdot π - \frac{π}{12} = \frac{12 π - 9 π - π}{12} = \frac{π}{6}$ $pi-3/4*pi-pi/12=(12pi-9pi-pi)/12=pi/6$
with the Theorem of sines weget

$a = \frac{27 \cdot sin (\frac{π}{6})}{sin (\frac{π}{12})}$ $a=(27*sin(pi/6))/(sin(pi/12))$
note that

$sin (\frac{π}{6}) = \frac{1}{2}$ $sin(pi/6)=1/2$

$sin (\frac{π}{12}) = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$ $sin(pi/12)=(sqrt(3)-1)/(2sqrt(2))$
Now we use the Formula
$A = \frac{1}{2} \cdot a \cdot b \cdot sin (γ)$ $A=1/2*a*b*sin(gamma)$

so we get

$A = \frac{1}{2} \cdot 27^{2} \cdot \frac{\sqrt{2}}{\sqrt{3} - 1}$ $A=1/2*27^2*sqrt(2)/(sqrt(3)-1)$
this is

$A = \frac{1}{4} \cdot 27^{2} \cdot (\sqrt{3} + 1)$ $A=1/4*27^2*(sqrt(3)+1)$

We have used that
$\frac{1}{\sqrt{3} - 1} = \frac{\sqrt{3} + 1}{2}$ $1/(sqrt(3)-1)=(sqrt(3)+1)/2$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 27, what is the area of the triangle?

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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is 3π4(3pi)/4 and the angle between sides B and C is π12pi/12. If side B has a length of 27, what is the area of the triangle?

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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 27, what is the area of the triangle?