Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $5$ $5$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $5$ $5$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-19T06:11:55

Longest possible perimeter of the triangle is

$P = a + b + c \approx 17.9538$ $color(brown)(P = a + b + c ~~ 17.9538$

Explanation:

To find the longest possible perimeter of the triangle.

Given $ˆ A = \frac{π}{3}, ˆ B = \frac{π}{4}$ $hatA = pi/3, hatB = pi/4$ , one $s i d e = 5$ $side = 5$

$ˆ C = π - \frac{π}{3} - \frac{π}{4} = \frac{5 π}{12}$ $hatC = pi - pi/3 - pi/4 = (5pi)/12$

Angle $ˆ B$ $hatB$ will correspond to side 5 to get the longest perimeter.

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$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a / sin A = b / sin B = c / sin C$ , applying sine law.

$a = \frac{b sin A}{sin B} = \frac{5 \cdot sin (\frac{π}{3})}{sin (\frac{π}{4})} = 6.1237$ $a = (b sin A) / sin B = (5 * sin (pi/3)) / sin (pi/4) = 6.1237$

$c = \frac{b sin C}{sin B} = \frac{5 \cdot sin (\frac{5 π}{12})}{sin (\frac{π}{4})} = 6.8301$ $c = (b sin C) / sin B = (5 * sin ((5pi)/12)) / sin (pi/4) = 6.8301$

Longest possible perimeter of the triangle is

$P = a + b + c = 6.1237 + 5 + 6.8301 \approx 17.9538$ $color(brown)(P = a + b + c = 6.1237 + 5 + 6.8301 ~~ 17.9538$

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $5$ $5$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Explanation:

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $5$ $5$ , what is the longest possible perimeter of the triangle?