A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-12-02T13:15:50

Largest possible area of the triangle $= 0.45$ $= color(blue)(0.45)$

Explanation:

Sum of the three angles of a triangle is equal to $180^{0} or π^{c}$ $180^0 or pi^c$

$∠ A = \frac{π}{12}, ∠ B = \frac{π}{8},$ $/_A = (pi)/12, /_B = pi/8,$
$∠ C = π - ((\frac{π}{12}) + (\frac{π}{8})) = π - \frac{5 π}{24} = \frac{19 π}{24}$ $/_C = pi -(( pi/12) + (pi/8)) =pi - (5pi)/24 = (19pi)/24$

To get the largest possible area, length 1 should correspond to the smallest $∠ A = \frac{π}{12}$ $/_A = pi/12$

$\frac{a}{sin (∠ A)} = \frac{b}{sin (∠ B)} = \frac{c}{sin (∠ C)}$ $a / sin( /_A )= b / sin( /_B )= c / sin( /_C)$

$\frac{1}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{π}{8})} = \frac{c}{sin (\frac{19 π}{24})}$ $1 / sin (pi/12) = b / sin (pi/8) = c / sin ((19pi)/24)$

$b = \frac{1 \cdot sin (\frac{π}{8})}{sin (\frac{π}{12})}$ $b = (1 * sin ((pi)/8)) / sin (pi /12)$
$b = \frac{0.3827}{0.2588} = 1.4787$ $b = 0.3827 / 0.2588 =color(blue)( 1.4787)$

$c = \frac{1 \cdot sin (19 \frac{π}{24})}{sin (\frac{π}{12})}$ $c = (1*sin (19pi/24))/ sin (pi/12)$
$b = \frac{0.6088}{0.2588} = 2.3524$ $b = 0.6088 / 0.2588 = color(blue)( 2.3524)$

Semi-Perimeter $s = \frac{a + b + c}{2} = \frac{1 + 1.4787 + 2.3524}{2} = 2.4156$ $s = (a + b + c )/2 =( 1 + 1.4787 + 2.3524 )/2= color (green)(2.4156)$

$s - a = 2.4156 - 1 = 1.4156$ $s - a = 2.4156 - 1 = 1.4156$
$s - b = 2.4156 - 1.4787 = 0.9369$ $s - b = 2.4156 - 1.4787 = 0.9369$
$s - c = 2.4156 - 2.3524 = 0.0632$ $s - c = 2.4156 - 2.3524 = 0.0632$

Area of $Delta ABC = sqrt(s (s-a) (s - b) (s - c))$
$= sqrt ( 2.4156 * 1.4156 * 0.9369 * 0.0632) = 0.45$

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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