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Difficulty: HardSine and Cosine Rules

In ΔABC\Delta ABC, side lengths are given as a=6 cma = 6\text{ cm} and b=63 cmb = 6\sqrt{3}\text{ cm}, and ∠A=30∘\angle A = 30^\circ. If ∠B\angle B is an obtuse angle, what is the length of side cc?

  1. 6 cm6\text{ cm}Answer
  2. B
    12 cm12\text{ cm}
  3. C
    63 cm6\sqrt{3}\text{ cm}
  4. D
    33 cm3\sqrt{3}\text{ cm}

Answer

The length of side cc is 6 cm6\text{ cm}.
Using the Sine Rule, 6sin⁡30∘=63sin⁡B\frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B}, which yields sin⁡B=32\sin B = \frac{\sqrt{3}}{2}. The two possible values for ∠B\angle B are 60∘60^\circ (acute) and 120∘120^\circ (obtuse). The problem explicitly specifies that ∠B\angle B is obtuse, so ∠B=120∘\angle B = 120^\circ. Subtracting from 180∘180^\circ gives ∠C=30∘\angle C = 30^\circ. Since ∠A=∠C=30∘\angle A = \angle C = 30^\circ, the triangle is isosceles, making side cc equal to side aa, which is 6 cm6\text{ cm}.

Step-by-Step Solution

1
Apply the Sine Rule to find sin⁡B\sin B
asin⁡A=bsin⁡B  ⟹  6sin⁡30∘=63sin⁡B  ⟹  12=63sin⁡B  ⟹  sin⁡B=32\frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B} \implies 12 = \frac{6\sqrt{3}}{\sin B} \implies \sin B = \frac{\sqrt{3}}{2}
The Sine Rule relates side lengths to the sines of their opposite angles.
2
Determine the value of ∠B\angle B using the given condition
∠B=60∘\angle B = 60^\circ or ∠B=180∘−60∘=120∘\angle B = 180^\circ - 60^\circ = 120^\circ. Since ∠B\angle B is obtuse, ∠B=120∘\angle B = 120^\circ.
The inverse sine function yields two possible angle solutions between 0∘0^\circ and 180∘180^\circ (the ambiguous case).
3
Calculate the third angle ∠C\angle C
∠C=180∘−(∠A+∠B)=180∘−(30∘+120∘)=30∘\angle C = 180^\circ - (\angle A + \angle B) = 180^\circ - (30^\circ + 120^\circ) = 30^\circ
The sum of interior angles in any triangle is 180∘180^\circ.
4
Find the length of side cc
Since ∠A=30∘\angle A = 30^\circ and ∠C=30∘\angle C = 30^\circ, ΔABC\Delta ABC is isosceles with side c=a=6 cmc = a = 6\text{ cm}.
Sides opposite to equal angles in a triangle are equal in length.

Key Concept

Ambiguous Case of the Sine Rule (SSA Condition)

Alternative Method

Alternatively, apply the Cosine Rule for angle AA: a2=b2+c2−2bccos⁡A  ⟹  62=(63)2+c2−2(63)ccos⁡30∘a^2 = b^2 + c^2 - 2bc \cos A \implies 6^2 = (6\sqrt{3})^2 + c^2 - 2(6\sqrt{3})c \cos 30^\circ. Simplifying gives 36=108+c2−18c  ⟹  c2−18c+72=036 = 108 + c^2 - 18c \implies c^2 - 18c + 72 = 0. Factoring yields (c−6)(c−12)=0(c - 6)(c - 12) = 0, giving c=6 cmc = 6\text{ cm} or c=12 cmc = 12\text{ cm}. For c=12 cmc = 12\text{ cm}, b2+a2=108+36=144=c2b^2 + a^2 = 108 + 36 = 144 = c^2, making ∠C=90∘\angle C = 90^\circ and ∠B=60∘\angle B = 60^\circ (acute). Thus, c=6 cmc = 6\text{ cm} corresponds to the obtuse angle ∠B=120∘\angle B = 120^\circ.
Estimated Time:2m 0s
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Topics

Sine and Cosine Rules

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