Question

Draw an equilateral triangle ABC and locate its centroid G. By measurements check whether if
(i) AB = BE = CF, (ii) GB = GC = GA

Answer 1

Answer:

Step-by-step explanation:

Construction: We have drawn median AD, BE and CF.

Let G be the centroid of triangle ABC.

Triangle ABC is an equilateral triangle.

∴ AB = BC = CA. And ∠ABC=∠BAC=∠BCA=60

And we can also say that AB/2=BC/2=CA/2 ⇒ BF =AF= BD = EC=AE.

Now, in △BFCand△BEC, we have:

BC= BC (Common Side)

∠FBC=∠ECB=60

BF = EC.

So, by SAS rule of congruence, we can write:

△BFC ≅△BEC.

∵ The two triangles are congruent. So, all angles and sides of one triangle are equal to corresponding angles and sides of another triangle.

Therefore , we have BE= CF. (1)

Now, in △ABEand△ABD, we have:

AB= AB (Common Side)

∠BAE=∠ABD=60

BD = AE.

So, by SAS rule of congruence, we can write:

△ABE≅△ABD

∵ The two triangles are congruent. So, all angles and sides of one triangle are equal to corresponding angles and sides of another triangle.

Therefore , we have BE= AD. (2)

From equation 1 and 2, we get:

AD = BE = CF. (3)

Now , we know that the G( the centroid) of the triangle divides the median in a 2:3 ratio.

So, on dividing the equation 4 by 2/3, we get:

2/3AD=2/3BE=2/3CF.

GA = GB = GC.

So, we can say that G is equidistant from the three vertices A. B and C.

Hence, it is proved that G is centroid as well as centre of circumcircle of the equilateral triangle.