A triangle has two corners with angles of $\frac{π}{4}$ $( pi ) / 4$ and $\frac{7 π}{12}$ $( 7 pi )/ 12$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{4}$ $( pi ) / 4$ and $\frac{7 π}{12}$ $( 7 pi )/ 12$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?

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Answer 1 · 2016-10-25T04:30:34

Largest possible area of the triangle is $33.467$ $33.467$

Explanation:

As two angles of a triangle are $\frac{π}{4}$ $pi/4$ and $\frac{7 π}{12}$ $(7pi)/12$ , the third angle is

$π - \frac{π}{4} - \frac{7 π}{12} = \frac{12 π - 3 π - 7 π}{12} = \frac{2 π}{12} = \frac{π}{6}$ $pi-pi/4-(7pi)/12=(12pi-3pi-7pi)/12=(2pi)/12=pi/6$

All triangles with these angles are similar. As length of one side is $7$ $7$ , its area will be maximum, if this side is opposite smallest angle i.e. $\frac{π}{6}$ $pi/6$ .

Area of a triangle given one side $a = 7$ $a=7$ and two angles $∠ A = \frac{π}{6}$ $/_A=pi/6$ , $∠ B = \frac{π}{4}$ $/_B=pi/4$ and $∠ C = \frac{7 π}{12}$ $/_C=(7pi)/12$ is

$\frac{a^{2} sin B sin C}{2 sin A}$ $(a^2sinBsinC)/(2sinA)$

= $\frac{7^{2} sin (\frac{7 π}{12}) sin (\frac{π}{4})}{2 sin (\frac{π}{6})}$ $(7^2sin((7pi)/12)sin(pi/4))/(2sin(pi/6))$

= $\frac{49 \times 0.9659 \times 0.7071}{2 \times 0.5}$ $(49xx0.9659xx0.7071)/(2xx0.5)$

= $\frac{49 \times 0.0 .683}{1}$ $(49xx0.0.683)/1$

= $33.467$ $33.467$

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A triangle has two corners with angles of $\frac{π}{4}$ $( pi ) / 4$ and $\frac{7 π}{12}$ $( 7 pi )/ 12$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of ( pi ) / 4 π4 ( pi ) / 4 and ( 7 pi )/ 12 7π12 ( 7 pi )/ 12 . If one side of the triangle has a length of 7 77 , what is the largest possible area of the triangle?

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Explanation:

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A triangle has two corners with angles of $\frac{π}{4}$ $( pi ) / 4$ and $\frac{7 π}{12}$ $( 7 pi )/ 12$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?