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The three angles of a triangle are proportional to 4: 1: 1. What is the ratio of the longest side to the total perimeter of the triangle?

1. 3/(2+√3)
2. 1/(2+√3)
3. √3/(√3+2)
4. 2/(2+√3)

Correct answer: √3/(√3+2)

Solution

The angles of the triangle are in the ratio 4:1:1, which means the angles measure 4x, x, and x. The longest side, opposite the largest angle, is proportional to the sine of that angle, leading to the ratio of the longest side to the perimeter being √3/(√3+2) when calculated using the sine rule.

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