Question

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio

• A. \(2:3:5\)
• B. \(1:2:3\)
• C. \(1:3:7\)
• D. \(3:7:9\)

Hint:

For equilateral triangle, \(R = 2r\) and \(r_1 = 3r\).

Answer 1

The Correct Option is B

To solve the problem, let us first understand the terms involved when dealing with an equilateral triangle:

1. In-radius (r): The radius of the circle inscribed within the triangle.
2. Circum-radius (R): The radius of the circle circumscribed around the triangle.
3. Ex-radius (r_e): The radius of the excircle opposite to any side of the triangle.

In an equilateral triangle with side length \(s\), these radii are given by:

• In-radius: \(r = \frac{s \sqrt{3}}{6}\)
• Circum-radius: \(R = \frac{s}{\sqrt{3}}\)
• Ex-radius: \(r_e = \frac{s \sqrt{3}}{2}\)

Next, let’s find the ratios:

1. Ratio of the in-radius to the circum-radius:
   • \(\frac{r}{R} = \frac{\frac{s \sqrt{3}}{6}}{\frac{s}{\sqrt{3}}} = \frac{1}{2}\)
2. Ratio of the circum-radius to the ex-radius:
   • \(\frac{R}{r_e} = \frac{\frac{s}{\sqrt{3}}}{\frac{s \sqrt{3}}{2}} = \frac{1}{3}\)

Now, combining these results, the ratio of the in-radius, circum-radius, and one of the ex-radii is:

• \(r:R:r_e = 1:2:3\)

This matches the given correct answer option: \(1:2:3\).

Therefore, the final answer is:

• In-radius, circum-radius, and one of the ex-radii of an equilateral triangle are in the ratio: \(1:2:3\).