The three interior angles of a triangle are in the ratio 1 : 1 : 4.\nIf the perimeter of the triangle is k times its largest side, what is the exact value of k?

Introduction / Context:
This problem mixes triangle angle ratios with side-perimeter relations. By converting the ratio of angles into concrete angle measures and then using the Law of Sines (or well-known values for special angles), we can express each side in terms of the largest side and hence obtain the perimeter-to-largest-side factor k.

Given Data / Assumptions:

• Angle ratio 1 : 1 : 4 implies angles 30°, 30°, and 120° (sum 180°).
• Largest angle is 120°, so the side opposite it is the largest side L.
• Perimeter P = k * L; we seek k.

Concept / Approach:
By the Law of Sines, side lengths are proportional to the sines of their opposite angles. Thus the two smaller sides are proportional to sin 30° = 1/2, while the largest side is proportional to sin 120° = sin 60° = √3/2. Therefore the ratio (smaller side):(largest side) = (1/2):(√3/2) = 1/√3.

Step-by-Step Solution:

Verification / Alternative check:
Assign a scale: let L correspond to √3/2. Then smaller sides are each 1/2. P = √3/2 + 1/2 + 1/2 = √3/2 + 1. Divide by L = √3/2 to get (√3/2 + 1)/(√3/2) = 1 + 2/√3, confirming the expression.

Why Other Options Are Wrong:

• 1 - 2/√3: Incorrect sign; perimeter must exceed L.
• 2 + 2/√3: Overcounts; would imply each smaller side equals L.
• 2: Would require sum of smaller sides = L, which is false here.

Common Pitfalls:

• Using sin 120° = −sin 60° (sign confusion); lengths use absolute values of sine.
• Assuming the triangle is acute; it is obtuse (120°).

Final Answer:
1 + 2/√3