Question

1. choose the law and the formula that would be used to solve the triangle.

law
formula
a. $a^2 = b^2 + c^2 - 2bc \cos a$
b. $c^2 = a^2 + b^2 - 2ab \cos c$
c. $b^2 = a^2 + c^2 - 2ac \cos b$
d. law of cosines
e. law of sines

Explanation:

Step1: Identify the given information

We have a triangle with two sides and the included angle: side \( a = 4.5 \, \text{cm} \), side \( c = 4 \, \text{cm} \), and included angle \( B = 84^\circ \). We need to find side \( b \).

Step2: Determine the appropriate law

The Law of Cosines is used when we have two sides and the included angle (SAS) to find the third side. The Law of Sines is used when we have a side - angle pair (AAS, ASA, or SSA). Since we have SAS (two sides and the included angle), we use the Law of Cosines.

Step3: Determine the appropriate formula from the Law of Cosines

The Law of Cosines formula is \( b^{2}=a^{2}+c^{2}-2ac\cos B \) when we want to find side \( b \) with sides \( a \), \( c \) and included angle \( B \). Looking at the options, option \( c \) is \( b^{2}=a^{2}+c^{2}-2ac\cos B \) and the law is the Law of Cosines (option \( d \)).

Answer:

Law: d. Law of Cosines
Formula: c. \( b^{2}=a^{2}+c^{2}-2ac\cos B \)