Two corners of a triangle have 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 $5$ $5$ , what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have 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 $5$ $5$ , what is the longest possible perimeter of the triangle?

1 Answer

Impact of this question

1356 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-13T18:00:33

The longest possible perimeter is, $p = 18.66$ $p = 18.66$

Explanation:

Let $∠ A = \frac{π}{6}$ $angle A = pi/6$

Let $∠ B = \frac{2 π}{3}$ $angle B = (2pi)/3$

Then $∠ C = π - ∠ A - ∠ B$ $angle C = pi - angle A - angle B$

$∠ C = π - \frac{π}{6} - \frac{2 π}{3}$ $angle C = pi - pi/6 - (2pi)/3$

$∠ C = \frac{π}{6}$ $angle C = pi/6$

To obtain the longest perimeter, we associate the given side with the smallest angle but we have two angles that are equal, therefore, we shall use the same length for both associated sides:

side $a = 5$ $a = 5$ and side $c = 5$ $c = 5$

We may use the Law of Cosines to find the length of side b:

$b = \sqrt{a^{2} + c^{2} - 2 (a) (c) cos (∠ B)}$ $b = sqrt(a^2 + c^2 - 2(a)(c)cos(angle B)$

$b = \sqrt{5^{2} + 5^{2} - 2 (5) (5) cos (\frac{2 π}{3})}$ $b = sqrt(5^2 + 5^2 - 2(5)(5)cos((2pi)/3)$

$b = 5 \sqrt{2 - 2 cos (\frac{2 π}{3})}$ $b = 5sqrt(2 - 2cos((2pi)/3)$

$b \approx 8.66$ $b ~~ 8.66$

The longest possible perimeter is, $p = 8.66 + 5 + 5 = 18.66$ $p = 8.66 + 5 + 5 = 18.66$

Science

Math

Humanities

... and beyond

Two corners of a triangle have 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 $5$ $5$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Two corners of a triangle have angles of 2π3 (2 pi )/ 3 and π6 ( pi ) / 6 . If one side of the triangle has a length of 5 5 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Two corners of a triangle have 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 $5$ $5$ , what is the longest possible perimeter of the triangle?