Question

How do you solve the right triangle ABC if b=120m, A=12?

Answer 1

`color(maroon)("Case 1 : "`
`color(indigo)(a = 24.95m,hatB = 90^@, hatC = 78^@, c = 117.38m, A_t = 1464.32 " sq m"`

`color(green)("Case 2 : "`
`color(green)(a = 25.51m, hatB = 78^@, hatC = 90^@, c = 122.68m, A_t = 1530.6 " sq m"`

Explanation:

Case 1 :

`b = 120 m, hat B = 90^@, hat A = 12^@`

`hat C = 180 - 90 - 12 = 78^@`

Applying the Law of Sines,

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

`a = (120 * sin 12) / sin 90 = 24.95 " m"`

`c = (120 * sin 78) / sin 90 = 117.38 " m"`

`"Area of rt triangle " A_t = (1/2) * a * c = (1/2) * 24.95 * 117.38 = color(indigo)(1464.32 " sq m"`

Case 2 :

`b = 120 m, hat A = 12^@, hat A = 90^@`

`hat B = 180 - 12 - 90 = 78^@`

`color(green)("Applying the Law of Sines,"`

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

`a = (b*sin A) / sin B = (120 * sin 12) / sin 78 = 25.51 " m"`

`"Applying Pythagoras theorem,"`

`c = sqrt(a^2 + b^2) = sqr(25.51^2 + 120^2) = 122.68 " m"`

`"Area of rt triangle "A_t = (1/2) * a * b = (1/2) * 25.51 * 120 = color(green)(1530.6 " sq m"`