Two corners of a triangle have angles of $\frac{7 π}{12}$ $(7 pi )/ 12$ and $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of 1, what is the longest possible perimeter of the triangle?

Question

1329 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-01-26T05:22:44

Longest possible perimeter of the triangle ABC is $P = 4.3461$ $color(green)(P = 4.3461)$

enter image source here
Given $A = \frac{7 π}{12}, B = \frac{π}{4}$ $A = (7pi)/12, B = pi/4$

Third angle $C = π - (\frac{7 π}{12} + \frac{π}{4}) = \frac{π}{6}$ $C = pi - ((7pi)/12 + pi/4) = pi/6$

To get the largest perimeter, side 1 to correspond to least angle $\frac{π}{6}$ $pi/6$
We know,

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a / sin A = b / sin B = c / sin C$

$\frac{1}{sin (\frac{π}{6})} = \frac{b}{sin (\frac{π}{4})} = \frac{c}{sin (\frac{7 π}{12})}$ $1/ sin (pi/6) = b / sin (pi/4) = c / sin ((7pi)/12)$

$b = \frac{1 \cdot sin (\frac{π}{4})}{sin (\frac{π}{6})} = 1.4142$ $b = (1 * sin (pi/4)) / sin (pi/6) = 1.4142$

$c = \frac{1 \cdot sin (\frac{7 π}{12})}{sin (\frac{π}{6})} = 1.9319$ $c = (1 * sin ((7pi)/12)) / sin (pi/6) = 1.9319$

Perimeter of triangle, $P = \frac{a + b + c}{2}$ $P = (a + b + c)/2$

$P = (1 + 1.4142 + 1.9319) = 4.3461$ $P = (1 + 1.4142 + 1.9319) = color(green)(4.3461)$

Two corners of a triangle have angles of 7π12 (7 pi )/ 12 and π4 pi / 4 . If one side of the triangle has a length of 1, what is the longest possible perimeter of the triangle?