A 30 60 90 Triangle with a long leg of 12 Find the length of the hypotenuse and the short leg.

Step-by-step explanation:   5 17 7 22

the long leg would be opposite the middle angle 60°,  the hypotenuse is opposite the 90° angle thus leaving the 30° is opposite the short leg

1) use sin Θ = opp / hyp       sin 60° = 12 / hyp    to find the hypotenuse

1) use sin Θ = opp / hyp       sin 30° = opp/ hyp    to find the short leg

      hyp = 12 / sin(60)       = 13.856  hypotenuse

      opp = (12 / sin(60)) (sin 30)     =  6.92   short leg

short leg ²  + long leg ²  = hypotenuse ²

     48         +     144        =     192          sum of the legs squared = hyp²

the value of x is 9 ⇒ answer b

step-by-step explanation:

* we can solve this problem using cosine and sine rule

- in δ abc

∵ ab = 36 , bc = 28 , ac = 16

- lets find the measure of angle b using the cosine rule

∵ cos b = [ab² + bc² - ac²]/2(ab)(bc)

∴ cos b = [36² + 28² - 16²]/2(36)(28)

∴ cos b = 19/21

∴ m∠b = cos^-1(19/21) ≅ 25°

- lets find the measure of angle a using the sine rule

∵ sina/bc = sinb/ac = sinc/ab

∴ sina/28 = sin25°/16 ⇒ by using cross multiplication

∴ sina = 28 × sin25°/16

∴ sina = 0.7396

∴ m∠a = sin^-1(0.7396) ≅ 48°

- from the figure m∠abd = m∠dbc

∵ m∠abc = 25°

∴ m∠abd = 25°/2 = 12.5°

- in δabd

∵ m∠a = 48° , m∠abd = 12.5°

∵ the sum of the measures of the interior angles of any δ is 180°

∴ m∠adb = 180° - (48° + 12.5°) = 119.5°

∵ ab = 36

∵ ad = x

- use the sine rule to find x

∵ sin∠adb/ab = sin∠abd/x

∴ sin119.5°/36 = sin12.5°/x ⇒ by using cross multiplication

∴ x = 36 × sin12.5°/sin119.5° = 8.95 ≅ 9

* the value of x is 9