A triangle has two corners with angles of $\frac{π}{2}$ $( pi ) / 2$ and $\frac{5 π}{12}$ $( 5 pi )/ 12$ . If one side of the triangle has a length of $14$ $14$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-04-01T07:26:12

$color(blue)("Largest possible area of the right ") color(crimson)(Delta = 365.75 " sq units"$

Given $hat A = pi/2, hat B = (5pi)/12, hat C = pi - pi/2 - (5pi)/12 = pi/12$

To get the largest possible areaof the right triangle,

side 14 should correspond to the least angle $hat C = pi/12$

Applying the law of sines,

$a / sin A = b / sin b = c / sin C$

$b = (c * sin B) / sin C$

$b = (14 * sin ((5pi)/12)) / sin (pi/12) = 52.25$

Hence $"Largest possible area of right "$

$Delta = (1/2) * b * c = (1/2) * 14 * 52.25 = 365.75 " sq units"$

A triangle has two corners with angles of ( pi ) / 2 π2 ( pi ) / 2 and ( 5 pi )/ 12 5π12 ( 5 pi )/ 12 . If one side of the triangle has a length of 14 1414 , what is the largest possible area of the triangle?