84. The lengths of two sides of a triangle are 2 units and 3 units and the angle included by these two sides is 60°. The length of the third side of the triangle will be
(A) √5 units
(B) √7 units
(C) 4 units
(D) 5 units
The Lengths of Two Sides of a Triangle are 2 Units and 3 Units and the Included Angle is 60°. Find the Length of the Third Side
Questions based on triangles are among the most frequently asked topics in competitive examinations such as CSIR NET, IIT JAM, GATE, CUET PG, and other entrance tests. Whenever two sides of a triangle and the angle between them are given, the most appropriate method is to apply the Law of Cosines, also known as the Cosine Rule. This theorem extends the Pythagorean theorem to all types of triangles and provides a direct relationship between the sides of a triangle and its included angle.
Correct Answer
Option (B) √7 units
Detailed Solution
Since two sides of the triangle and the angle included between them are known, the Law of Cosines is the most suitable formula to determine the unknown side. Let the unknown side be c.
The Law of Cosines is given by
c² = a² + b² − 2ab cos θ
Here,
a = 2 units
b = 3 units
θ = 60°
Substituting these values into the formula,
c² = 2² + 3² − 2 × 2 × 3 × cos 60°
Since
cos 60° = 1/2
Therefore,
c² = 4 + 9 − 12 × (1/2)
c² = 13 − 6
c² = 7
Taking the positive square root,
c = √7 units
Hence, the length of the third side of the triangle is √7 units.
Option-wise Explanation
Option (A) √5 units
This value does not satisfy the Law of Cosines for the given sides and included angle. Therefore, this option is incorrect.
Option (B) √7 units
Using the Cosine Rule, the third side is obtained as √7 units. Hence, this is the correct answer.
Option (C) 4 units
If the third side were 4 units, substituting it into the Cosine Rule would not satisfy the given angle of 60°. Therefore, this option is incorrect.
Option (D) 5 units
A third side of 5 units would correspond to a completely different triangle and does not satisfy the given conditions. Hence, this option is incorrect.
Why the Law of Cosines is Used
The Pythagorean theorem is applicable only to right-angled triangles, whereas the given triangle has an included angle of 60°. The Law of Cosines is a generalized formula that works for every triangle, making it the ideal choice whenever two sides and the included angle are known.
Concept Behind the Question
The Law of Cosines establishes a relationship between the lengths of the three sides of a triangle and the cosine of one of its angles. It is one of the most important formulas in geometry and trigonometry because it enables the calculation of an unknown side without first determining any other angle. This concept is widely used in mathematics, physics, engineering, navigation, and competitive examinations.
Final Answer
Length of the third side = √7 units
Correct Option: (B)
