The ratio of AO + BO + CO to the perimeter of the triangle is 1/√3
Since O is the circumcenter of the equilateral triangle ABC, OA = OB = OC = l.
Also, since the triangle is an equilateral triangle, its sides are equal. So, AB = BC = AC = a.
Since the base angles of the equilateral triangle are 60°, OA, OB and AB form an isosceles triangle with base angles 30° since OA and OB bisect the base angles.
So ∠AOB + ∠OAB + ∠OBA = 180°
∠AOB + 30° + 30° = 180° (∠OAB = ∠OBA = 30°)
∠AOB + 60° = 180°
∠AOB = 180° - 60°
∠AOB = 120°
Using the sine rule in triangle AOB,
OA/sinOBA = AB/sinAOB
l/sin30° = a/sin120°
l = asin30°/sin120°
l = a × 1/2 ÷ √3/2
l = a/√3
Since AO = BO = CO = l,
AO + BO + CO = l + l + l
= 3l
= 3a/√3
Also the perimeter of the triangle P = AB + BC + AC. Since AB = BC = AC,
P = AB + BC + AC
= a + a + a
= 3a
So, the ratio of AO + BO + CO to the perimeter of the triangle is 3a/√3 ÷ 3a = 1/√3
Thus, the ratio of AO + BO + CO to the perimeter of the triangle is 1/√3
Learn more about equilateral triangles here:
https://brainly.com/question/17757837
