Introduction

Finding the third side of a triangle with sides 2 and 3 and angle 60 is a classic exercise in applying the Law of Cosines, a powerful tool for non‑right triangles.

This article walks through the complete solution, checks every multiple‑choice option, and explains clearly why √7 units is the correct answer.

Step‑by‑step solution using Law of Cosines

For a triangle with sides a, b, and c, and included angle C between sides a and b, the Law of Cosines is

c² = a² + b² − 2ab cos C

where c is the side opposite angle C.

Here:

• a = 2 units
• b = 3 units
• C = 60°

Compute squares of the known sides:

• a² = 2² = 4
• b² = 3² = 9

Use cos 60° = 1/2.

Substitute into the formula:


c² = 4 + 9 − 2·2·3·cos 60°
c² = 13 − 12·1/2 = 13 − 6 = 7


Take the square root:

c = √7 units

Hence, the length of the third side is √7 units, matching option (B).

Explanation of each option

Option (A) √5 units

If the third side were √5, then the Law of Cosines run in reverse would give:

cos C = (a² + b² − c²) / (2ab) = (4 + 9 − 5) / (2·2·3) = 8 / 12 = 2/3

This would mean C = cos⁻¹(2/3), which is not 60°, so √5 cannot be correct.

Option (B) √7 units

Using √7 in the cosine formula:

cos C = (4 + 9 − 7) / (2·2·3) = 6 / 12 = 1/2

This matches cos 60° = 1/2, so √7 is fully consistent with the given data, making this the correct option.

Option (C) 4 units

Assuming the third side is 4:

cos C = (4 + 9 − 16) / (2·2·3) = −3 / 12 = −1/4

The angle would then satisfy C = cos⁻¹(−1/4), which is an obtuse angle greater than 90°, not 60°, therefore 4 units does not fit the condition.

Option (D) 5 units

With third side 5:

cos C = (4 + 9 − 25) / (2·2·3) = −12 / 12 = −1

This implies C = 180°, which cannot be the interior angle of a non‑degenerate triangle with sides 2, 3, and 5, so 5 units is impossible in this context.

Key takeaway for similar problems

• When given two sides and the included angle (SAS), use the Law of Cosines to find the third side of any triangle.
• Always check that the resulting side length and angle values are consistent with basic triangle properties, such as feasible angle ranges and the triangle inequality.