Q6 of 13 Page 96

Let’s prove that the length of the medians are equal in length in an equilateral triangle.


Let an equilateral ΔABC with three medians AE, BF and CG


So, E, F and G are the midpoints of BC , AC and AB


∴ BE=EC, AF=FC and AG=GB


AB=BC=AC


⇒ � AB= � BC= � AC


⇒ AG=BE=AF


Now, in ΔAEC and ΔBFA,


EC=AF


AC=BC (given)


∠ACB=∠BAC (each 60° )


∴ ΔAEC ≅ ΔBFA (by SAS rule)


So, AE=BF (by cpct)………(1)


Similarly, ΔCGB≅ ΔBFA (by SAS rule)


So, CG=BF (by cpct)……….(2)


And, ΔCGB≅ ΔAEC


So, CG=AE…………..(3)


From (1),(2) and (3),


AE=BF=CG


Hence, proved.


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