A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $18$ $18$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $18$ $18$ , what is the largest possible area of the triangle?

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Answer 1 · 2016-06-22T07:12:39

Largest possible area of triangle is $280.584$ $280.584$

Explanation:

The third angle of the triangle is $π - \frac{π}{3} - \frac{π}{6} = \frac{6 π}{6} - \frac{2 π}{6} - \frac{π}{6} = \frac{3 π}{6} = \frac{π}{2}$ $pi-pi/3-pi/6=(6pi)/6-(2pi)/6-pi/6=(3pi)/6=pi/2$ .

Such a triangle will have the largest possible area if the side with length of $18$ $18$ is opposite smallest angle $\frac{π}{6}$ $pi/6$ . Let the other two sides be $b$ $b$ and $c$ $c$ . Then according to sine formula, we have

$\frac{18}{sin (\frac{π}{6})} = \frac{b}{sin (\frac{π}{3})} = \frac{c}{sin (\frac{π}{2})}$ $18/sin(pi/6)=b/sin(pi/3)=c/sin(pi/2)$ or

$\frac{18}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{3}}{2}} = \frac{c}{1}$ $18/(1/2)=b/(sqrt3/2)=c/1$ or $36 = \frac{b}{\frac{\sqrt{3}}{2}} = c$ $36=b/(sqrt3/2)=c$ and hence

$c = 36$ $c=36$ and $b = 36 \times \frac{\sqrt{3}}{2} = 18 \sqrt{3}$ $b=36xxsqrt3/2=18sqrt3$

As it is a right angled triangle and side opposite right angle is $c$ $c$ ,

area of triangle is given by $18 \times 18 \sqrt{3} \frac{)}{2} = 162 \times \sqrt{3}$ $18xx18sqrt3)/2=162xxsqrt3$

= $162 \times 1.732 = 280.584$ $162xx1.732=280.584$

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A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $18$ $18$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of π3 ( pi ) / 3 and π6 ( pi )/ 6 . If one side of the triangle has a length of 1818 , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $18$ $18$ , what is the largest possible area of the triangle?