Given a triangle with sides $a$, $b$, and $c$, and angle $B$ opposite side $b$, which equation can be used to find the value of $b$ if side $a$ measures 8.7 cm?

• A. $b^2=a^2+c^2-2accos\left(C\right)$
• B. $b^2=a^2+c^2-2accos\left(A\right)$
• C. $b^2={8.7}^2+c^2+28.7ccos\left(B\right)$
• D. $b^2={8.7}^2+c^2-28.7ccos\left(B\right)$

Answer: (D)

Verified step by step guidance

1. Identify the given elements: side \( a = 8.7 \) cm, side \( b \) opposite angle \( B \), and side \( c \). We want to find an equation to calculate \( b \).
2. Recall the Law of Cosines formula, which relates the sides and angles of a triangle: \[ b^{2} = a^{2} + c^{2} - 2ac \cos(B) \] This formula is used when you know two sides and the included angle or want to find a side opposite a known angle.
3. Substitute the known value \( a = 8.7 \) into the Law of Cosines formula: \[ b^{2} = (8.7)^{2} + c^{2} - 2 \times 8.7 \times c \times \cos(B) \] This expresses \( b^{2} \) in terms of \( c \) and angle \( B \).
4. Note that the angle used in the cosine term must be the angle opposite the side \( b \), which is angle \( B \). Using any other angle (like \( A \) or \( C \)) would be incorrect for finding \( b \).
5. Therefore, the correct equation to find \( b \) when \( a = 8.7 \) cm is: \[ b^{2} = (8.7)^{2} + c^{2} - 2 \times 8.7 \times c \times \cos(B) \] This equation can be used to solve for \( b \) once \( c \) and \( B \) are known.