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
2 answers
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
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
12 answers