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Difficulty: MediumSine and Cosine Rules

In ΔPQR\Delta PQR, side p=4 cmp = 4\text{ cm}, side q=42 cmq = 4\sqrt{2}\text{ cm}, and ∠P=30∘\angle P = 30^\circ. If ∠Q\angle Q is an obtuse angle, what is the measure of ∠Q\angle Q?

  1. 135∘135^\circAnswer
  2. B
    45∘45^\circ
  3. C
    60∘60^\circ
  4. D
    150∘150^\circ

Answer

The measure of angle QQ is 135∘135^\circ.
Applying the Sine Rule gives sin⁡Q=42⋅sin⁡30∘4=22\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \frac{\sqrt{2}}{2}. The acute reference angle is 45∘45^\circ, so the obtuse angle is 180∘−45∘=135∘180^\circ - 45^\circ = 135^\circ.

Step-by-Step Solution

1
Apply the Sine Rule to relate sides p,qp, q and angles P,QP, Q
psin⁡P=qsin⁡Q  ⟹  4sin⁡30∘=42sin⁡Q\frac{p}{\sin P} = \frac{q}{\sin Q} \implies \frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin Q}
The Sine Rule connects pairs of opposite sides and angles in any triangle.
2
Solve for sin⁡Q\sin Q
sin⁡Q=42⋅sin⁡30∘4=2⋅12=22\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}
Substitute sin⁡30∘=12\sin 30^\circ = \frac{1}{2} and simplify the numerical expression.
3
Determine the obtuse solution for angle QQ
Q=180∘−45∘=135∘Q = 180^\circ - 45^\circ = 135^\circ
Since sin⁡Q=22\sin Q = \frac{\sqrt{2}}{2}, the principal acute angle is 45∘45^\circ. Because the problem specifies that angle QQ is obtuse (90∘<Q<180∘90^\circ < Q < 180^\circ), we take its supplement.

Key Concept

Ambiguous case of the Sine Rule (SSA condition)
Estimated Time:1m 30s
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Topics

Sine and Cosine Rules

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