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

Introduction / Context:
This problem links angle ratios in a triangle with the ratio of perimeter to the largest side. It tests knowledge of how triangle side lengths relate to angles via trigonometric functions, especially in non right triangles. Recognizing that side lengths are proportional to sines of opposite angles allows us to express perimeter in terms of the largest side and then determine the factor k.

Given Data / Assumptions:

• Angles of the triangle are in ratio 1 : 1 : 4.
• Let the three angles be A, B, and C with C being the largest angle.
• Perimeter P is k times the largest side.
• We are asked to find k.
• All trigonometric values are taken in degrees, and the triangle is valid in Euclidean geometry.

Concept / Approach:
If the angles are in the ratio 1 : 1 : 4, we can set A = x, B = x, and C = 4x. Because A + B + C = 180°, we get 6x = 180°, so x = 30°. Thus the angles are 30°, 30°, and 120°, with 120° as the largest angle. In any triangle, side lengths are proportional to the sines of the opposite angles. Let the largest side, opposite the 120° angle, be denoted by a. Then the other two sides have lengths proportional to sin 30°. Using these proportions, we can express the perimeter in terms of a and simplify to get k.

Step-by-Step Solution:
Let angles be A = 30°, B = 30°, C = 120°.Let side a be opposite angle C = 120°, and sides b and c be opposite angles A and B respectively.By sine rule, a : b : c = sin 120° : sin 30° : sin 30°.Compute sin 120° = sin 60° = √3 / 2 and sin 30° = 1 / 2, so a : b : c = (√3 / 2) : (1 / 2) : (1 / 2) = √3 : 1 : 1.Let a = √3 units, then b = 1 and c = 1. Perimeter P = a + b + c = √3 + 1 + 1 = √3 + 2. Since P = k * a, we have k = (√3 + 2) / √3 = 1 + 2 / √3.

Verification / Alternative check:
To verify, compute k numerically. With √3 ≈ 1.732, we have P ≈ 1.732 + 2 = 3.732. The largest side a ≈ 1.732. Thus k ≈ 3.732 / 1.732 ≈ 2.155. Computing 1 + 2 / √3 gives 1 + 2 / 1.732 ≈ 1 + 1.155 ≈ 2.155, which matches. This confirms that the expression 1 + 2 / √3 correctly represents the ratio of perimeter to the largest side.

Why Other Options Are Wrong:
Option 1 − 2 / √3 would yield a number less than 1 and cannot represent the ratio of perimeter to the largest side, since perimeter is always larger than any individual side. Option 2 + 2 / √3 is too large and does not match the numeric check. Option 2 corresponds roughly to P ≈ 2a, which is not correct for this specific triangle. Option None of these is not needed because one of the given expressions matches the correct ratio.

Common Pitfalls:
Some students mistakenly treat the angle ratio as side ratio without using trigonometric relations. Others forget that the largest angle corresponds to the largest side and instead assign labels inconsistently. Errors also occur when computing sine values, especially for 120°, or when simplifying the algebraic expression for k. Careful use of the sine rule and correct substitution prevents such mistakes.

Final Answer:
The value of k is 1 + 2/√3.