Question

on which triangle can the law of cosines be used to find the length of an unknown side? law of cosines: ( a^2 = b^2 + c^2 - 2bc cos(a) )

Explanation:

To determine which triangle the Law of Cosines can be applied to, we recall the Law of Cosines formula: \( a^2 = b^2 + c^2 - 2bc \cos(A) \), which is used when we know two sides and the included angle (SAS) or all three sides (SSS).

Analyzing Each Triangle:

1. First Triangle (Top - Left):

• Sides: \( QS = 12 \), \( RS = 7 \); Angle at \( S \): \( 57^\circ \).
• This is a SAS case (two sides \( QS = 12 \), \( RS = 7 \), and included angle \( 57^\circ \) at \( S \)).
• The Law of Cosines can be used here to find the unknown side \( QR \) (or \( q \), depending on labeling).

1. Second Triangle (Top - Middle):

• Angles: \( \angle Q = 36^\circ \), \( \angle S = 57^\circ \), \( \angle R = 87^\circ \); Side \( QS = 12 \).
• This is a triangle with all angles known and one side (AAS/ASA). The Law of Cosines is not needed here (Law of Sines is more appropriate for AAS/ASA).

1. Third Triangle (Top - Right):

• Angles: \( \angle Q = 36^\circ \), \( \angle S = 57^\circ \); Side \( QS = 12 \).
• Only two angles and one side are given (AAS/ASA). Law of Sines is more appropriate, not Law of Cosines.

Conclusion:

The Law of Cosines applies to the first triangle (top - left) because it has two known sides and the included angle (SAS), matching the requirements for the Law of Cosines.

Answer:

The triangle with sides \( QS = 12 \), \( RS = 7 \), and \( \angle S = 57^\circ \) (top - left triangle).