A triangle has two corners with angles of $\frac{2 π}{3}$ $(2 pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . 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 $\frac{2 π}{3}$ $(2 pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . 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 · 2018-01-20T14:47:41

Largest possible triangle will have sides $7, 7, 12.1244$ $color(brown)(7, 7, 12.1244)$

Smallest possible triangle will have sides $7, 4.0415, 4.0415$ $color(red)(7, 4.0415, 4.0415)$

Explanation:

The three angles are $\frac{2 π}{3}, \frac{π}{6}, \frac{π}{6}$ $(2pi)/3, pi/6, pi/6$ as sumof the three angles equals $π^{c}$ $pi^c$ . It’s an isosceles triangle as two angles measure $\frac{π}{6}$ $pi/6$ each.

Case 1 : To get largest triangle possible

Side 7 should correspond to the smallest angle $\frac{π}{6}$ $pi/6$

$:.7 / sin (pi/6) = a / sin ((2pi)/3)$

$a = (7 * sin ((2pi)/3)) / sin (pi/6) ~~ 12.1244$

Three sides are $color(brown)(7, 7, 12.1244)$

Case 2 : To get smallest triangle possible

Side 7 should correspond to the largest angle $((2pi)/3)$

$7 / sin ((2pi)/3) = b / sin (pi/6)$

$b = (7 * sin (pi/6)) / sin ((2pi)/3) = 4.0415$

Three sides are $color(red)(7, 4.0415, 4.0415)$

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A triangle has two corners with angles of $\frac{2 π}{3}$ $(2 pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . 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 (2 pi ) / 3 2π3 (2 pi ) / 3 and ( pi )/ 6 π6 ( pi )/ 6 . 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 $\frac{2 π}{3}$ $(2 pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?