The sides of a triangle are in the ratio 1 : √3 : 2, then the angles of the triangle are in the rati

Q: Why do we introduce the constant $k$ if it just cancels out anyway?

Great question! While it does cancel, introducing $k$ is a mark of a disciplined physicist. It ensures dimensional consistency. In more complex problems, forgetting the scale factor can lead to errors. It is a habit that saves you from 'silly' mistakes.

Q: Can I use the Sine Rule instead?

Absolutely! You could use $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. However, the Cosine Rule is often more robust when you have all three sides. It gives you the cosine value directly, which is unambiguous for angles between $0^\circ$ and $180^\circ$.

Q: Is there a shortcut to identify this triangle?

Yes! If you notice that $1^2 + (\sqrt{3})^2 = 2^2$, you immediately recognize the Pythagorean theorem $a^2 + b^2 = c^2$. This tells you the triangle is right-angled at $C$. Recognizing this early saves time, but the Cosine Rule method is the rigorous way to prove it.

Question

The sides of a triangle are in the ratio $1 : \sqrt{3} : 2$, then the angles of the triangle are in the ratio

Options:

(A) $1 : 3 : 5$

(B) $2 : 3 : 4$

(C) $3 : 2 : 1$

(D) $1 : 2 : 3$

Sigma Percentile · Accepted Answer

### The Geometry of Ratios: Unlocking the Triangle

Welcome, future engineers! Today, we are not just solving a problem; we are uncovering the hidden architecture of a triangle.

When you see a ratio like $1 : \sqrt{3} : 2$, your intuition should immediately start buzzing. This is not just any triangle; it is a classic signature in geometry. But before we jump to conclusions, let us walk through the rigorous path to prove it.

### Phase 1: The Setup

We begin by defining our sides. We are given the ratio $a : b : c = 1 : \sqrt{3} : 2$.

To work with these as physical lengths, we introduce a constant of proportionality, $k$. This is a vital step in your JEE journey—always maintain dimensional consistency.

So, we define our sides as $a = k$, $b = \sqrt{3}k$, and $c = 2k$. Imagine these sides laid out before you. Our mission is to find the angles $A$, $B$, and $C$ that correspond to these sides.

### Phase 2: The Bridge (The Cosine Rule)

How do we connect side lengths to angles? We need a bridge. That bridge is the **Cosine Rule**.

It is a powerful, elegant tool that relates the sides of any triangle to the cosine of one of its angles. For angle $A$, the formula is:

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Notice the beauty of this equation: the side opposite to the angle, $a$, is the one being subtracted. It is a perfect balance.

### Phase 3: The Calculation

Now, let us substitute our values. We replace $a$, $b$, and $c$ with our $k$-expressions:

$$\cos A = \frac{(\sqrt{3}k)^2 + (2k)^2 - (k)^2}{2(\sqrt{3}k)(2k)}$$

Take a breath and expand carefully. The numerator becomes $3k^2 + 4k^2 - k^2$, which simplifies to $6k^2$. The denominator becomes $4\sqrt{3}k^2$.

Look at that! The $k^2$ terms cancel out perfectly, leaving us with $\cos A = \frac{6}{4\sqrt{3}}$. Simplifying this, we get $\frac{3}{2\sqrt{3}}$, which is simply $\frac{\sqrt{3}}{2}$.

We know from our trigonometric tables that $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Thus, **$A = 30^\circ$**. We have our first angle!

### Phase 4: The Conclusion

We apply the same logic for angle $B$:

$$\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{k^2 + 4k^2 - 3k^2}{4k^2} = \frac{2k^2}{4k^2} = \frac{1}{2}$$

Since $\cos B = \frac{1}{2}$, we find **$B = 60^\circ$**.

With $A = 30^\circ$ and $B = 60^\circ$, we use the fundamental property that the sum of angles in a triangle is $180^\circ$. Therefore, **$C = 180^\circ - (30^\circ + 60^\circ) = 90^\circ$**.

We have discovered a right-angled triangle! The ratio of the angles is $30^\circ : 60^\circ : 90^\circ$, which simplifies beautifully to $1 : 2 : 3$. You have successfully decoded the geometry. Keep this analytical mindset, and no problem will ever be too complex for you.

Animated Solution for Mathematics - Trigonometry: The sides of a triangle are in the ratio $1 : 3 : 2$ , then the angles of the triangle are in the ratio

Select Answer:

Visualized Solution

Visualizing the Triangle

The Cosine Rule for Angle $A$

Substituting Values for $cos A$

Simplifying the Expression

Calculating Angle $A$

The Cosine Rule for Angle $B$

Calculating Angle $B$

Finding the Third Angle $C$

Final Ratio of Angles

The Sigma Insight: Properties of Triangles

The Geometry of Ratios

Phase 1

Phase 2

Phase 3

Phase 4

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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; } The sides of a triangle are in the ratio 1:3​:2, then the angles of the triangle are in the ratio