A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

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Answer 1 · 2016-06-26T08:46:00

21.85252

Explanation:

Given for $△ A B C$ $triangle ABC$ , $∠ B = \frac{π}{4}, ∠ C = \frac{π}{3}$ $/_B = pi/4, /_C=pi/3$ .

We know that sum of the 3 angles of a triangle is $π$ $pi$ . Hence $∠ A = π - \frac{π}{3} - \frac{π}{4}$ $/_A = pi - pi/3 - pi/4$ => $∠ A = \frac{5 π}{12}$ $/_A = (5pi)/12$

As the angles $∠ B < ∠ C < ∠ A$ $/_B < /_C < /_A$ , length of side b < side c < side A

To get the largest possible area of the triangle with any one side with 8, we must have the shortest side as 8.

Then b must be 8

$\frac{b}{sin B}$ $b/Sin B$ = $\frac{8}{sin (\frac{π}{4})}$ $8/Sin (pi/4)$ = $\frac{8}{\frac{\sqrt{2}}{2}}$ $8/(sqrt 2/2)$ = $\frac{16}{\sqrt{2}}$ $16/sqrt 2$ = $8 \sqrt{2}$ $8 sqrt 2$

Using the law of Sines,

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

a = $(\frac{b}{sin B}) \cdot sin A$ $(b/Sin B) * Sin A$ = $8 \sqrt{2}$ $8 sqrt 2$ * $sin (\frac{5 π}{12})$ $sin ((5pi)/12)$ =

c= $(\frac{b}{sin B}) \cdot sin C$ $(b/Sin B) * Sin C$ = $8 \sqrt{2}$ $8 sqrt 2$ * $sin (\frac{π}{3})$ $sin ((pi)/3)$

Area of the triangle = $\cdot \frac{a \cdot b \cdot sin C}{2}$ $*(a*b*Sin C)/2$

= $8 \sqrt{2}$ $8 sqrt 2$ * $sin (\frac{5 π}{12})$ $sin ((5pi)/12)$ 8 $sin (\frac{π}{3})$ $sin ((pi)/3)$

= 21.85252

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?