A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{3 π}{8}$ $(3 pi )/ 8$ . If one side of the triangle has a length of $7$ $7$ , 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 $\frac{3 π}{8}$ $(3 pi )/ 8$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-12-02T13:37:24

Largest possible area of the triangle $= 29.5747$ $= color(blue)(29.5747)$

Explanation:

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

$∠ A = \frac{π}{4} ∠ B = 3 \frac{π}{8},$ $/_A = (pi)/4 /_B = 3pi/8,$
$∠ C = π - ((\frac{π}{4}) + (3 \frac{π}{8})) = π - (5 \frac{π}{8}) = \frac{3 π}{8}$ $/_C = pi -(( pi/4) + (3pi/8)) =pi - (5pi/8) = (3pi)/8$

To get the largest possible area, length 1 should correspond to the smallest $∠ A = \frac{π}{4}$ $/_A = pi/4$

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

$\frac{7}{sin (\frac{π}{4})} = \frac{b}{sin (\frac{3 π}{8})} = \frac{c}{sin (\frac{3 π}{8})}$ $7 / sin (pi/4) = b / sin ((3pi)/8) = c / sin ((3pi)/8)$

$b = \frac{7 \cdot \frac{sin (3 π)}{8}}{sin (\frac{π}{4})}$ $b = (7 * sin (3pi)/8) / sin (pi /4)$
$b = \frac{6.4672}{0.7071} = 9.1461$ $b = 6.4672 / 0.7071 =color(blue)(9.1461)$

$c = \frac{7 \cdot \frac{sin (3 π)}{8}}{sin (\frac{π}{4})}$ $c = (7*sin (3pi)/8) / sin (pi/4)$
$c = 9.1461$ $c = color(blue)( 9.1461)$

Semi-Perimeter $s = \frac{a + b + c}{2} = \frac{7 + 9.1461 + 9.1461}{2} = 12.6461$ $s = (a + b + c )/2 =( 7 + 9.1461 + 9.1461)/2 = color (green)(12.6461)$

$s - a = 12.6461 - 7 = 5.6461$ $s - a = 12.6461 - 7 = 5.6461$
$s - b = 12.6461 - 9.1461 = 3.5$ $s - b = 12.6461 - 9.1461 = 3.5$
$s - c = 12.6461 - 9.1461 = 3.5$ $s - c = 12.6461 - 9.1461 = 3.5$

Area of $Delta ABC = sqrt(s (s-a) (s - b) (s - c))$
$= sqrt ( 12.6461 * 5.6461 * 3.5 * 3.5) = 29.5747$

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{3 π}{8}$ $(3 pi )/ 8$ . If one side of the triangle has a length of $7$ $7$ , 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 $\frac{3 π}{8}$ $(3 pi )/ 8$ . If one side of the triangle has a length of $7$ $7$ , what is the largest possible area of the triangle?