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Difficulty: EasyLaw of Sines and Law of Cosines

In triangle XYZXYZ, the length of side XYXY is 1414 meters, the measure of ∠X\angle X is 40∘40^\circ, and the measure of ∠Z\angle Z is 80∘80^\circ. Which of the following expressions represents the length, in meters, of side YZYZ?

  1. 14sin⁡(40∘)sin⁡(80∘)\frac{14 \sin(40^\circ)}{\sin(80^\circ)}Answer
  2. B
    14sin⁡(80∘)sin⁡(40∘)\frac{14 \sin(80^\circ)}{\sin(40^\circ)}
  3. C
    sin⁡(40∘)14sin⁡(80∘)\frac{\sin(40^\circ)}{14 \sin(80^\circ)}
  4. D
    14sin⁡(40∘)sin⁡(80∘)14 \sin(40^\circ) \sin(80^\circ)
  5. E
    14sin⁡(40∘)sin⁡(80∘)\frac{14}{\sin(40^\circ) \sin(80^\circ)}

Answer

14sin⁡(40∘)sin⁡(80∘)\frac{14 \sin(40^\circ)}{\sin(80^\circ)}
The expression derived by applying the Law of Sines asin⁡(A)=csin⁡(C)\frac{a}{\sin(A)} = \frac{c}{\sin(C)} correctly matches side YZYZ with its opposite angle ∠X=40∘\angle X = 40^\circ and side XY=14XY = 14 with its opposite angle ∠Z=80∘\angle Z = 80^\circ, yielding YZ=14sin⁡(40∘)sin⁡(80∘)YZ = \frac{14 \sin(40^\circ)}{\sin(80^\circ)}.

Step-by-Step Solution

1
Identify the relevant law and relate the known sides and angles
Using the Law of Sines: YZsin⁡(X)=XYsin⁡(Z)\frac{YZ}{\sin(X)} = \frac{XY}{\sin(Z)}
The Law of Sines relates the side lengths of a triangle to the sines of their opposite angles.
2
Substitute the given values into the formula
YZsin⁡(40∘)=14sin⁡(80∘)\frac{YZ}{\sin(40^\circ)} = \frac{14}{\sin(80^\circ)}
Side XY=14XY = 14 is opposite ∠Z=80∘\angle Z = 80^\circ, and side YZYZ is opposite ∠X=40∘\angle X = 40^\circ.
3
Solve for the unknown side YZYZ
YZ=14sin⁡(40∘)sin⁡(80∘)YZ = \frac{14 \sin(40^\circ)}{\sin(80^\circ)}
Multiply both sides of the equation by sin⁡(40∘)\sin(40^\circ) to isolate YZYZ.

Key Concept

Law of Sines
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Topics

Law of Sines and Law of Cosines

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