Let $Δ A B C$ be a triangle whose area is $103$ units with side lengths $∣ A B ∣ = 8$ units and $∣ A C ∣ = 5$ units. Find possible values of the angle A.

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $60^{\circ}$ or $120^{\circ}$

Detailed Solution

Area $= \frac{1}{2} \cdot A B \cdot A C \cdot s in A = \frac{1}{2} \cdot 8 \cdot 5 \cdot s in A = 20 s in A$ .

$20 s in A = 103$ , so $s in A = \frac{3}{2}$ .

In a triangle $0^{\circ} < A < 180^{\circ}$ , so $A = 60^{\circ}$ or $A = 120^{\circ}$ .

✓ Check

s in 120^{\circ} = s in 60^{\circ} = \frac{3}{2}

, so both angles give area

20 \cdot \frac{3}{2} = 103

⚠ Common trap

(C)

30^{\circ}

only comes from using

cos A = \frac{3}{2}

instead of

s in A

Quick tip

$Δ = \frac{1}{2} b c sin A \Rightarrow sin A = \frac{3}{2} \Rightarrow A = 6 0^{\circ} or 12 0^{\circ}$

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Properties of Triangles — Preparation

This question tests your understanding of Properties of Triangles, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Properties of triangles connect the sides a, b, c and the opposite angles A, B, C of any triangle, not just right-angled ones. This topic covers the sine rule (a/sin A = b/sin B = c/sin C = 2R), the cosine rule (a^2 = b^2 + c^2 - 2bc cos A), the projection formulas, half-angle formulas, and formulas for the area of a triangle, including (1/2)ab sin C and Hero's formula. It also covers the circumradius R, the inradius r and their relationships with the area and semi-perimeter, such as r = area/s. Exam questions test applying these rules to find an unknown side, angle or area, and identifying the type of triangle from a given condition. Choosing between the sine rule and the cosine rule is usually the key decision. Practise the properties of triangles questions below, each with a step-by-step solution.

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Let $Δ A B C$ be a triangle whose area is $103$ units with side lengths $∣ A B ∣ = 8$ units and $∣ A C ∣ = 5$ units. Find possible values of the angle A.

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Let ΔABC be a triangle whose area is 103​ units with side lengths ∣AB∣=8 units and ∣AC∣=5 units. Find possible values of the angle A.

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Properties of Triangles — Preparation

Let $Δ A B C$ be a triangle whose area is $103$ units with side lengths $∣ A B ∣ = 8$ units and $∣ A C ∣ = 5$ units. Find possible values of the angle A.