A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $3$ $3$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $3$ $3$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-02-27T16:11:35

Largest possible area of the triangle is

$A_{t} = 10.864$ $A_t = color(indigo)(10.864$ sq units

Explanation:

$ˆ A = \frac{π}{2}, ˆ B = \frac{π}{8}$ $hat A = pi/2, hat B = pi/8$

Third angle $ˆ C = π - \frac{π}{8} / \frac{- π}{2} = \frac{3 π}{8}$ $hat C = pi - pi / 8 /- pi /2 = (3pi)/8$

To get largest area, side with length 3 should correspond to the smallest angle $\frac{π}{8}$ $pi/8$

Applying law of sines,

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$\frac{3}{sin (\frac{π}{8})} = \frac{b}{sin (\frac{π}{2})} = \frac{c}{sin (3 π)} / 8$ $3 / sin (pi/8) = b / sin (pi/2) = c / sin (3pi)/8$

$b = \frac{3 \cdot sin (\frac{π}{2})}{sin (\frac{π}{8})} = 7.8394$ $b = (3 * sin (pi/2)) / sin (pi/8) = 7.8394$

Area of the tan be arrived at by using the formula

$A_{t} = (\frac{1}{2}) a b sin C = (\frac{1}{2}) \cdot 3 \cdot 7.8394 \cdot sin (\frac{3 π}{8}) = 10.864$ $A_t = (1/2) a b sin C = (1/2) * 3 * 7.8394 * sin ((3pi)/8) = 10.864$ sq units.

Alternatively,

$c = \frac{3 \cdot sin (\frac{3 π}{8})}{sin (\frac{π}{8})} = 7.2426$ $c = (3 * sin ((3pi)/8)) / sin (pi / 8) = 7.2426$

Since it’s a right triangle,

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$A_{t} = (\frac{1}{2}) a c = (\frac{1}{2}) 3 \cdot 7.2426 = 10.864$ $A_t = (1/2) a c = (1/2) 3 * 7.2426 = 10.864$

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $3$ $3$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $3$ $3$ , what is the largest possible area of the triangle?