Animated Solution for Mathematics - Trigonometry: In a triangle $ABC$, $a:b:c=4:5:6$. The ratio of the radius of the circumcircle to that of the incircle is ..................

Unveiling the Ratio of Radii

Welcome, future engineers. Today, we are not just solving a problem; we are exploring the intrinsic harmony of a triangle. When you look at a triangle with side ratios $4:5:6$, you are looking at a specific geometric signature.

This ratio defines the triangle's shape, and today, we will uncover the elegant relationship between its two most important circles: the incircle and the circumcircle.

The Geometry of Proportions

We begin with the given ratio $a:b:c=4:5:6$. To work with these lengths, we introduce a scaling factor $k$.

Thus, our sides are $a=4k$, $b=5k$, and $c=6k$. This $k$ is our bridge between the abstract ratio and the concrete geometry.

Our goal is to find the ratio $\frac{R}{r}$, where $R$ is the circumradius and $r$ is the inradius. Imagine this triangle in your mind—it is a scalene triangle, and we are about to dissect its internal structure.

The Semi-perimeter and Heron's Bridge

To find the radii, we must first understand the triangle's area. The most powerful tool in our arsenal for a triangle with known sides is Heron's formula.

First, we calculate the semi-perimeter $s$:

Now, we prepare the components for Heron's formula: $s-a$, $s-b$, and $s-c$. These are the building blocks of our area calculation:

Take a moment to appreciate these values. They are the 'distances' from the semi-perimeter to each side, and they are essential for the area $S=\sqrt{s(s-a)(s-b)(s-c)}$.

Computing the Area

Let us assemble the pieces into Heron's formula:

Multiplying the numerators, we get $15\cdot 7\cdot 5\cdot 3=1575$. The $k$ terms multiply to $k^4$, and the denominator is $2\cdot 2\cdot 2\cdot 2=16$.

Simplifying this, we find:

This area is the heart of our triangle's geometry.

The Radii and the Final Ratio

Now, we define our radii. The inradius $r$ is the radius of the circle inscribed within the triangle, touching all sides. Its relationship to the area is $r=\frac{S}{s}$:

Next, the circumradius $R$, the radius of the circle passing through all three vertices, is given by $R=\frac{abc}{4S}$:

Finally, we find the ratio $\frac{R}{r}$:

The result, $16:7$, is a testament to the consistency of geometry. The scaling factor $k$ vanished, proving that for any triangle with sides in the ratio $4:5:6$, the ratio of the circumradius to the inradius is always $16:7$.

You have successfully navigated the relationship between sides, area, and radii. Keep this logic in your toolkit—it is the foundation of many complex problems you will face in your journey to becoming an engineer.