find measurement of b,c, and ab

Question

Sol1 : m∠B =  °, m∠C =  °, AB = 

Sol2 : m∠B =  °, m∠C =  °, AB = 

Sol3 : m∠B =  °, m∠C =  °, AB = 

Answer 1

find measurement of b,c, and ab

angles: α=? opposite side a

β=? " " b

γ=80° " " c

sides: a=? shortestα

b=14 mid β

c=31 hypotenuse

a/sinα=b/sinβ=c/sinγ

14/sin?=31/sin80=31.478

14/31.478=.4447asin=26.4°=β; so α=73.6° mistakes.

To get side "a"; b^2+c^2-2bc(cosα)=a^2

196+961-(868)cos73.6=a^2

1157-253.778=a^2

903.22^.5=30.05°=α

The drawing is wrong. α would have to be over 90° making it c. I don't like 14.

Answer 2

We use the law of Sines to determine the missing angles and lengths.

a/sin(A) = b/sin(B) = c/sin(C)

We have A = 80, a = 31, and b = 14

31/sin(80) = 14/sin(B) ⇒ sin(B) =[(14)sin(80)]/31 = 0.4448

sin^-10.4448 = 26.41° = B

Now that we know the angle B, we can determine angle C. The sum of the angles of a triangle is 180°

180 - (80 + 26.41) = 73.59°

The last thing to do is calculate c

c/sin(73.59) = 31/sin(80)

c = (31)sin(73.59)/sin(80) = 30.19