The sides of a triangle are 7 cm, 5 cm and 3 cm. Then the largest angle of the triangle is:

Appeared in: HPSSC - 2015

Explanation

The largest angle in any triangle is always opposite the longest side. In this triangle, the longest side is 7 cm.
The Law of Cosines is the correct formula to use for finding an angle when all three side lengths are known. The formula is c² = a² + b² - 2ab cos(C).
By substituting the side lengths (a=5, b=3, c=7) into the formula, we get 7² = 5² + 3² - 2(5)(3)cos(C).
Solving the equation step-by-step: 49 = 34 - 30cos(C), which simplifies to 15 = -30cos(C).
This gives cos(C) = -15/30 = -0.5.
The angle C for which the cosine is -0.5 is 120°.

Option A: An angle of 45° would mean cos(C) is approximately 0.707, which does not match the calculated value of -0.5.
Option B: An angle of 60° would mean cos(C) is 0.5. The calculation resulted in -0.5, so this is incorrect.
Option C: An angle of 90° would indicate a right-angled triangle, where the Pythagorean theorem (a² + b² = c²) must hold true. Here, 5² + 3² = 34, which is not equal to 7² = 49.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of the Law of Cosines to find an angle in a triangle given three sides as background academic context rather than a clinical decision trigger.
This question tests general intelligence and mathematical aptitude, which are essential components of nursing entrance examinations.
Strong problem-solving and calculation skills are necessary for various nursing tasks, such as medication dosage calculations, interpreting lab results, and understanding physiological parameters.
What if the sides were 3, 4, and 5 cm? In that case, 3² + 4² = 9 + 16 = 25, which equals 5². The Pythagorean theorem holds, so the triangle would be a right-angled triangle, and the largest angle would be 90°.

How to Approach the Question

First, identify the information given: the lengths of all three sides of a triangle (7 cm, 5 cm, 3 cm).
Next, determine what the question is asking for: the measure of the largest angle.
Recall the geometric principle that the largest angle in a triangle is always opposite the longest side. Here, the longest side is 7 cm.
Select the appropriate mathematical tool. When you know all three sides and want to find an angle, the Law of Cosines (c² = a² + b² - 2ab cos(C)) is the correct formula.
Assign the variables correctly: let 'c' be the longest side (7 cm), and 'a' and 'b' be the other two sides (5 cm and 3 cm).
Substitute these values into the formula and carefully solve the equation for cos(C).

Concept Tested & Keywords

Concept Tested: Application of the Law of Cosines to find an angle in a triangle given three sides.
Stem keywords: triangle, sides, 7 cm, 5 cm, 3 cm, largest angle
Lead-in keywords: BEST, MOST RELEVANT CLUE

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Practise the full HPSSC - 2015

Attempt every question from this paper in a timed mock, then review the full solution for each one.