Question

A triangle has two corners with angles of `( pi ) / 2` and `( pi )/ 6`. If one side of the triangle has a length of `7`, what is the largest possible area of the triangle?

Answer 1

`Area of` largest possible `Delta = color (purple)(42.44)`

Explanation:

Three angles are `pi/2, (pi/6, (pi -( (pi/2)+ ((pi)/6)) =( pi/3`

`a/ sin A = b / sin B = c / sin C`

To get the largest possible are, smallest angle should correspond to the side of length 7

`7 / sin (pi/6) = b / sin ((pi)/2) = c / sin ((pi)/3)`

`b = (7*sin (pi/2)) / sin (pi / 6) = (7*1) / (1/2) = 14`

`c = (7* sin (pi/3)) / sin (pi/6) = (7 * (sqrt3/2))/(1/2) = 7 sqrt 3 = 12.1245`

Semi perimeter `s = (a + b + c) / 2 = (7 + 14 + 12.1245)/2 = 16.5623`

`s-a = 16.5623-7 = 9.5623`
`s-b = 16.5623-14 = 2.5623`
`s-c = 16.5623 - 12.1245 = 4.4378`

`Area of Delta = sqrt (s (s-a) (s-b) (s-c))`

`Area of Delta = sqrt( 16.5623 * 9.5623 * 2.5623 * 4.4378)`
`Area of` largest possible `Delta = color (purple)(42.44)`

Alternate method to calculate `Area of Delta`
Since it’s a right triangle,
`Area of Delta = (1/2) Base * ht = (1/2) * 7 * 12.1245 = color (purple)(42.44)`