Question

In an equilateral triangle, prove that the centroid and the circumcentre of triangle coincide. In the chapter circle of class 9​

Answer 1

Answer:

proved : The centroid and circumcentre are coincident.

Step-by-step explanation:

Given : An equilateral triangle ABC in which D, E and F are the mid- points of sides BC,

CA and AB respectively.

Construction : Draw medians, AD, BE and CF.

Proof : Let G be the centroid of ΔABC i. e., the point of intersection of AD, BE and CF. In triangles BEC and BFC, we have

∠B = ∠C = 60

BC = CE [∵ AB = AC ⇒ ½ AB = ½ AC ⇒ BF = CE]

∴ ΔBEC ≅ ΔBFC

⇒ BE = CF ……(i)

Similarly, ΔCAF ≅ ΔCAD

CF = AD …….(ii)

From (i) & (ii) AD = BE = CF

⇒ 2/3 AD = 2/3, BE = 2/3 CF

CG = 2/3 CF

GA = 2/3 AD, GB = 2/3 BE

⇒ GA = GB = GC

⇒ G is equidistant from the vertices

⇒ G is the circumcentre of ΔABC

Hence, the centroid and circumcentre are coincident.