In an equilateral triangle, (circumradius) : (inradius) : (exradius) is equal to,

A. 1:1:1

B. 1:2:3

C. 2:1:3

D. 3:2:4

To determine the ratio of the circumradius, inradius, and exradius of an equilateral triangle, we first need to understand what each of these terms means and how they relate to the triangle's dimensions. Let's break it down step by step.

Definitions of Key Terms

In an equilateral triangle:

• Circumradius (R): This is the radius of the circumcircle, the circle that passes through all three vertices of the triangle.
• Inradius (r): This is the radius of the incircle, the circle that is tangent to all three sides of the triangle.
• Exradius (r_ex): This is the radius of the excircle, which is tangent to one side of the triangle and the extensions of the other two sides. There are three exradii, one for each vertex.

Formulas for an Equilateral Triangle

For an equilateral triangle with side length 'a', the formulas for the circumradius, inradius, and exradius are as follows:

• Circumradius (R): R = a / √3
• Inradius (r): r = a / (2√3)
• Exradius (r_ex): r_ex = a / (2√3) (same as inradius, but for the opposite side)

Calculating the Ratios

Now, let's calculate the ratios of these radii:

• For circumradius: R = a / √3
• For inradius: r = a / (2√3)
• For exradius: r_ex = a / (2√3)

To find the ratios, we can express each radius in terms of the circumradius (R):

• Inradius: r = (1/2)R
• Exradius: r_ex = (1/2)R

Establishing the Ratio

Now we can express the circumradius, inradius, and exradius in a ratio:

• R : r : r_ex = R : (1/2)R : (1/2)R

This simplifies to:

• 1 : 1/2 : 1/2

To eliminate the fractions, we can multiply through by 2:

• 2 : 1 : 1

Final Ratio

However, we need to express the exradius in relation to the circumradius. Since the exradius is also equal to the inradius for an equilateral triangle, we can conclude that:

• R : r : r_ex = 1 : 1 : 1

Thus, the correct answer is option A: 1:1:1. This means that in an equilateral triangle, the circumradius, inradius, and exradius are all equal, reflecting the triangle's symmetrical properties.