Question:
If the angles of a triangle are in the ratio 4: 1: 1, then the ratio between its largest side and its perimeter is
If the angles of a triangle are in the ratio 4: 1: 1, then the ratio between its largest side and its perimeter is
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The Law of Sines (\( a = 2R~sin~A \)) allows you to instantly map geometric side ratios directly into simple trigonometric evaluations, completely bypassing the need to compute actual structural dimensions.
Updated On: Jun 8, 2026
- \( 1:(1+\sqrt{3}) \)
- \( 2:3 \)
- \( \sqrt{3}:(2+\sqrt{3}) \)
- \( 1:(2+\sqrt{3}) \)
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The Correct Option is C
Solution and Explanation
Concept:
Let the angles of the triangle be \( 4k, 1k, 1k \). Since the sum of angles in a triangle is \( 180^{\circ} \):
\[
4k + 1k + 1k = 180^{\circ} \implies 6k = 180^{\circ} \implies k = 30^{\circ}
\]
Thus, the three angles are \( A = 120^{\circ} \), \( B = 30^{\circ} \), and \( C = 30^{\circ} \).
According to the Law of Sines, side lengths are directly proportional to the sines of their opposite angles: \( a : b : c = sin~A : sin~B : sin~C \).
Step 1: Evaluating side length proportions.
\[ a : b : c = sin(120^{\circ}) : sin(30^{\circ}) : sin(30^{\circ}) \] \[ = \frac{\sqrt{3}}{2} : \frac{1}{2} : \frac{1}{2} = \sqrt{3} : 1 : 1 \] We can set the sides as \( a = \sqrt{3}x \), \( b = 1x \), and \( c = 1x \).
Step 2: Determining largest side and perimeter values.
The largest side is opposite the largest angle (\( 120^{\circ} \)), which is: \[ \text{Largest side} = a = \sqrt{3}x \] The total perimeter of the triangle is the sum of all three sides: \[ \text{Perimeter} = a + b + c = \sqrt{3}x + 1x + 1x = (2 + \sqrt{3})x \]
Step 3: Calculating the required ratio.
\[ \text{Ratio} = \frac{\text{Largest side}}{\text{Perimeter}} = \frac{\sqrt{3}x}{(2+\sqrt{3})x} = \frac{\sqrt{3}}{2+\sqrt{3}} \]
Step 1: Evaluating side length proportions.
\[ a : b : c = sin(120^{\circ}) : sin(30^{\circ}) : sin(30^{\circ}) \] \[ = \frac{\sqrt{3}}{2} : \frac{1}{2} : \frac{1}{2} = \sqrt{3} : 1 : 1 \] We can set the sides as \( a = \sqrt{3}x \), \( b = 1x \), and \( c = 1x \).
Step 2: Determining largest side and perimeter values.
The largest side is opposite the largest angle (\( 120^{\circ} \)), which is: \[ \text{Largest side} = a = \sqrt{3}x \] The total perimeter of the triangle is the sum of all three sides: \[ \text{Perimeter} = a + b + c = \sqrt{3}x + 1x + 1x = (2 + \sqrt{3})x \]
Step 3: Calculating the required ratio.
\[ \text{Ratio} = \frac{\text{Largest side}}{\text{Perimeter}} = \frac{\sqrt{3}x}{(2+\sqrt{3})x} = \frac{\sqrt{3}}{2+\sqrt{3}} \]
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