Question

question 5 of 10
in order to apply the law of sines to find the length of the side of a triangle, it is enough to know which of the following?
a. two sides and one angle
b. the area of the triangle
c. all three angles
d. two angles and a side

Explanation:

The Law of Sines states that for a triangle with sides \(a\), \(b\), \(c\) and opposite angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). To find a side using the Law of Sines, we need a known ratio of a side and its opposite angle. If we know two angles and a side, we can find the third angle (since the sum of angles in a triangle is \(180^\circ\)) and then use the Law of Sines to find the unknown side.

• Option A: Two sides and one angle may be the case for the Law of Cosines (if the angle is included) or ambiguous for Law of Sines (SSA case), not sufficient on its own for Law of Sines to find a side directly.
• Option B: The area of the triangle doesn't directly relate to the Law of Sines for finding a side length.
• Option C: Knowing all three angles only gives the shape (similar triangles), but no side length to set the scale, so we can't find the actual side length.
• Option D: Two angles and a side gives us a known side - angle pair (from the two angles, we can find the third angle and have a side with its opposite angle or use the known side and angle to find others), so we can apply the Law of Sines.

Answer:

D. Two angles and a side