A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{11 π}{24}$ $(11pi)/24$ . What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{11 π}{24}$ $(11pi)/24$ . What is the area of the triangle?

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Answer 1 · 2016-01-15T17:38:42

First of all let me denote the sides with small letters $a$ $a$ , $b$ $b$ and $c$ $c$ .
Let me name the angle between side $a$ $a$ and $b$ $b$ by $∠ C$ $/_ C$ , angle between side $b$ $b$ and $c$ $c$ by $∠ A$ $/_ A$ and angle between side $c$ $c$ and $a$ $a$ by $∠ B$ $/_ B$ .

Note:- the sign $∠$ $/_$ is read as "angle".
We are given with $∠ B$ $/_B$ and $∠ A$ $/_A$ . We can calculate $∠ C$ $/_C$ by using the fact that the sum of any triangles' interior angels is $π$ $pi$ radian.
$\Rightarrow ∠ A + ∠ B + ∠ C = π$ $implies /_A+/_B+/_C=pi$
$\Rightarrow \frac{11 π}{24} + \frac{11 π}{24} + ∠ C = π$ $implies (11pi)/24+(11pi)/24+/_C=pi$
$\Rightarrow ∠ C = π - (\frac{11 π}{24} + \frac{11 π}{24}) = π - \frac{11 π}{12} = \frac{π}{12}$ $implies/_C=pi-((11pi)/24+(11pi)/24)=pi-(11pi)/12=pi/12$
$\Rightarrow ∠ C = \frac{π}{12}$ $implies /_C=pi/12$

It is given that side $a = 7$ $a=7$ and side $b = 2 .$ $b=2.$

Area is also given by
$A r e a = \frac{1}{2} a \cdot b sin ∠ C$ $Area=1/2a*bSin/_C$

$\Rightarrow A r e a = \frac{1}{2} \cdot 7 \cdot 2 sin (\frac{π}{12}) = 7 \cdot 0.2588 = 1.8116$ $implies Area=1/2*7*2Sin(pi/12)=7*0.2588=1.8116$ square units
$\Rightarrow A r e a = 1.8116$ $implies Area=1.8116$ square units

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A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{11 π}{24}$ $(11pi)/24$ . What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is (11pi)/2411π24(11pi)/24 and the angle between B and C is (11pi)/2411π24 (11pi)/24. What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{11 π}{24}$ $(11pi)/24$ . What is the area of the triangle?