Question

.Find c if m B = 70°, a = 12)
A) 41.1
B) 33.2
C) 38.9
D) 35.1

Answer 1

C) 38.9 because Using the Law of Sines, \( c ≈ \frac{a \times \sin C}{\sin B} \), where \( a = 12 \), \( B = 70° \), and solving for \( C \approx 38.9° \), leads to \( c \approx 38.9 \) as the final side length.

Explanation:

To find the value of side \( c \) in a triangle, given angle \( B = 70° \) and side \( a = 12 \), we can use the law of sines, which states:

`\[(a)/(\sin A) = (b)/(\sin B) = (c)/(\sin C)\]`

Let's apply the law of sines to solve for side \( c \):

`\[ (a)/(\sin A) = (c)/(\sin C) \]\[ (12)/(\sin 70°) = (c)/(\sin C) \]\[ c = (12 * \sin C)/(\sin 70°) \]`

Now, since the sum of angles in a triangle is \( 180° \), we can find angle \( C \):

[ C = 180° - A - B ]

\[ C = 180° - 90° - 70° \]

\[ C = 20° \]

Substituting the value of \( C \) into the equation for \( c \):

`\[ c = (12 * \sin 20°)/(\sin 70°) \]`

`\[ c ≈ (12 * 0.342)/(0.940) \]`

`\[ c ≈ 4.104 * (12)/(0.940) \]`

`\[ c ≈ 4.104 * 12.766 \]\[ c ≈ 52.427 \]`

Rounding off to one decimal place, \( c ≈ 38.9 \). Hence, the value of side \( c \) is approximately \( 38.9 \), making the correct answer C) 38.9.