What is the ratio measures of the in-radius, circum-radius and one of the ex-radius of an equilateral triangle?
(1) 1 : 2 : 5
(2) 1 : 3 : 5
(3) 1 : 2 : 3****
(4) 1 : 1.4142 : 2
correct
Let a is the side of the equilateral triangle.
Then, Area of Δ =(√3/4)*a^2
s = (a+a+a)/2 = 3*a/2
Now, Inradius (r) = Area of Δ/s = (√3/4*a^2)/(3*a/2) = a/(2*√3)
Circumradius(R) = (a*b*c)/4*(Area of Δ) = a^3/(√3/a^2) = a/√3
Exradii (r1) = (Area of Δ)/(s−a)=(√3/4*a^2)/(a/2) = √3*a/2
∴ r : R : r1 = a/(2*√3) : a/√3 : √3*a/2 = 1:2:3
To find the ratio measures of the in-radius, circum-radius, and one of the ex-radii of an equilateral triangle, we can make use of some formulas and properties.
Let's begin by defining the terms:
- In-radius: The in-radius of a triangle is the radius of the inscribed circle, which touches all three sides of the triangle.
- Circum-radius: The circum-radius of a triangle is the radius of the circumcircle, which passes through all three vertices of the triangle.
- Ex-radius: The ex-radius of a triangle refers to the radius of the excircle, which is tangent to one side of the triangle and the extensions of the other two sides.
For an equilateral triangle, these measures are related by the following ratios:
In-radius : Circum-radius : One of the Ex-radii = 1 : 2 : 3
Therefore, the correct answer is option (3) 1 : 2 : 3.