If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the pe

Q: Wait, why do we assume the triangle is isosceles?

Great observation! In the ratio $4:1:1$, two of the parts are identical ($1$ and $1$). Since the angles are proportional to these parts, two angles must be equal. By definition, a triangle with two equal angles is an isosceles triangle. This symmetry is a huge gift in geometry!

Q: Why is the longest side $a$?

In any triangle, the side opposite the largest angle is always the longest. Since our angles are $120^\circ, 30^\circ,$ and $30^\circ$, the largest angle is $120^\circ$. Therefore, the side opposite this angle (which we labeled $a$) must be the longest side.

Q: Can I use the Cosine Rule instead of the Sine Rule?

You absolutely can! The Cosine Rule is a powerful tool. However, for this specific problem, the Sine Rule is much more elegant because it directly relates the ratios of the sides to the sines of the angles, saving you from calculating squares and square roots unnecessarily.

Question

If the angles of a triangle are in the ratio $4 : 1 : 1$, then the ratio of the longest side to the perimeter is

Options:

(A) $\sqrt{3} : (2 + \sqrt{3})$

(B) $1 : 6$

(C) $1 : 2 + \sqrt{3}$

(D) $2 : 3$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the mathematical landscape. Today, we are not just solving a problem; we are uncovering the hidden architecture of a triangle.

We are given a triangle where the angles exist in a ratio of $4 : 1 : 1$. At first glance, this might seem like a simple arithmetic exercise, but beneath the surface lies a beautiful, symmetric structure waiting to be revealed.

### The Foundation of Angles

Every triangle is bound by the sacred law of the Angle Sum Property: the sum of its interior angles must always be $180^{\circ}$. We are told the angles $A, B,$ and $C$ are in the ratio $4 : 1 : 1$.

Let us introduce a common multiplier, $x$. This allows us to define our angles as $4x, x,$ and $x$. By substituting these into our fundamental law, we get the equation:

$$4x + x + x = 180^{\circ}$$

Combining these terms, we find $6x = 180^{\circ}$, which leads us to $x = 30^{\circ}$.

Just like that, the mystery unfolds. Our angles are $120^{\circ}, 30^{\circ},$ and $30^{\circ}$. Because two angles are $30^{\circ}$, we are dealing with an obtuse isosceles triangle.

### Bridging Angles to Sides with the Sine Rule

Now that we know the angles, we use the Sine Rule to find the relationship between the sides. It states that the ratio of a side to the sine of its opposite angle is constant:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Substituting our angles, we have:

$$\frac{a}{\sin 120^{\circ}} = \frac{b}{\sin 30^{\circ}} = \frac{c}{\sin 30^{\circ}}$$

Recall your trigonometric values: $\sin 120^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$. Plugging these in, we get:

$$\frac{a}{\frac{\sqrt{3}}{2}} = \frac{b}{\frac{1}{2}} = \frac{c}{\frac{1}{2}}$$

By multiplying everything by $2$, we simplify this to the ratio $a : b : c = \sqrt{3} : 1 : 1$. This represents the 'DNA' of our triangle's side lengths.

### Final Calculation

The question asks for the ratio of the longest side to the perimeter. The longest side is opposite the largest angle, $120^{\circ}$, so our longest side is $a$, which is proportional to $\sqrt{3}$.

The perimeter $P$ is the sum of the sides:

$$P = a + b + c = \sqrt{3} + 1 + 1 = 2 + \sqrt{3}$$

Finally, we calculate the ratio $\frac{a}{P}$:

$$\frac{a}{P} = \frac{\sqrt{3}}{2 + \sqrt{3}}$$

To rationalize the denominator, we multiply the numerator and denominator by $(2 - \sqrt{3})$:

$$\frac{\sqrt{3}(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2\sqrt{3} - 3}{4 - 3} = 2\sqrt{3} - 3$$

The final ratio is **$2\sqrt{3} - 3$**.

Animated Solution for Mathematics - Trigonometry: If the angles of a triangle are in the ratio $4 : 1 : 1$ , then the ratio of the longest side to the perimeter is

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Visualized Solution

Visualizing the Triangle

Angle Sum Property

Setting up the Equation

Solving for $x$

Finding the Angles

Introducing the Sine Rule

Substituting Angle Values

Evaluating Sine Values

Simplifying Side Ratios

Identifying the Longest Side

Calculating the Perimeter

Simplifying the Perimeter

Final Ratio Calculation

The Sigma Insight: Properties of Triangles

Analyzing the Setup

The Foundation of Angles

Bridging Angles to Sides with the Sine Rule

Final Calculation

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the angles of a triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is