In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and

Q: Why is the distance between the centers of the two bottom coins equal to $2r$?

Because the two coins are touching each other, the distance between their centers is simply the sum of their radii, which is $r + r = 2r$. Since $r = 1$, this distance is $2$.

Q: Why is the angle $30^\circ$?

In an equilateral triangle, each angle is $60^\circ$. The line from the vertex to the center of the circle is the angle bisector, which splits the $60^\circ$ angle into two $30^\circ$ angles. This is a standard property of circles inscribed in polygons.

Q: Can I use the area of the triangle as $\frac{1}{2} \times \text{base} \times \text{height}$?

Yes, you can, but you would first need to calculate the height of the triangle, which is more complex than using the standard formula $\text{Area} = \frac{\sqrt{3}}{4} a^2$. Both methods will lead to the same result, but the formula is more direct.

Question

In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

Options:

(A) $4 + 2\sqrt{3}$

(B) $6 + 4\sqrt{3}$

(C) $12 + \frac{7\sqrt{3}}{4}$

(D) $3 + \frac{7\sqrt{3}}{4}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are not just solving a geometry problem; we are uncovering the hidden architecture of a beautiful arrangement. Imagine you are standing before an equilateral triangle, and inside it, three identical coins of radius $r=1$ are resting in perfect harmony.

They touch each other, and they touch the walls of the triangle. It is a scene of perfect symmetry. Our mission is to find the area of this triangle by first unlocking the secret of its side length, $a$.

### The Anatomy of the Side Length

When you look at the base of the triangle, the side length is not merely the diameter of the coins. There is a gap between the corner of the triangle and the point where the coin touches the base.

If we want to find the total side length $a$, we must break it down into three distinct segments. We have a segment from the left vertex to the first point of tangency, let's call this $x$. Then, we have the distance between the centers of the two bottom coins, which is $2r$.

Finally, we have another segment from the second point of tangency to the right vertex, which is also $x$. Thus, our total side length is:

$$a = x + 2r + x = 2x + 2r$$

### The Power of the Angle Bisector

Now, how do we find $x$? This is where the magic of geometry comes in. Focus on the bottom-left corner.

If you draw a line from the vertex to the center of the coin, you are drawing an angle bisector. Because the triangle is equilateral, every corner angle is $60^{\circ}$. The bisector cuts this in half, giving us a $30^{\circ}$ angle.

Now, drop a perpendicular from the center of the coin to the base. This is the radius $r = 1$. We have just created a right-angled triangle.

### Trigonometry to the Rescue

In this right-angled triangle, we have an angle of $30^{\circ}$, an opposite side of length $r = 1$, and an adjacent side of length $x$. Using the tangent ratio, we have:

$$	an(30^{\circ}) = \frac{	ext{Opposite}}{	ext{Adjacent}} = \frac{r}{x}$$

Substituting our known values, we get:

$$\frac{1}{\sqrt{3}} = \frac{1}{x}$$

Solving for $x$, we find $x = \sqrt{3}$. This is the missing piece of our puzzle.

### Final Calculation

Now that we have $x = \sqrt{3}$ and $r = 1$, we can find the total side length:

$$a = 2x + 2r = 2\sqrt{3} + 2$$

With $a$ in hand, we turn to the area formula for an equilateral triangle:

$$	ext{Area} = \frac{\sqrt{3}}{4} a^2$$

Substituting our value for $a$:

$$	ext{Area} = \frac{\sqrt{3}}{4} (2 + 2\sqrt{3})^2$$

Expanding the square, we calculate:

$$(2 + 2\sqrt{3})^2 = 4 + 12 + 8\sqrt{3} = 16 + 8\sqrt{3}$$

Finally, multiplying by $\frac{\sqrt{3}}{4}$:

$$	ext{Area} = \frac{\sqrt{3}}{4} (16 + 8\sqrt{3}) = 4\sqrt{3} + 6$$

The area of the triangle is **$6 + 4\sqrt{3}$** square units. You have successfully navigated the geometry and arrived at the solution. Keep this technique of breaking down complex shapes into right triangles in your toolkit; it will serve you well in your JEE journey!

Animated Solution for Mathematics - Trigonometry: In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

Select Answer:

Visualized Solution

The Setup

The Goal: Side Length $a$

Corner Geometry

The $3 0^{\circ}$ Angle

The Right Triangle

Trigonometry for $x$

Solving for $x$

The Middle Segment

Total Side Length $a$

Area Formula

Substituting $a$

Expanding the Square

Final Simplification

Conclusion

The Sigma Insight: Properties of Triangles

Analyzing the Setup

The Anatomy of the Side Length

The Power of the Angle Bisector

Trigonometry to the Rescue

Final Calculation

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