Question

Hence, AABC is isosceles, EXAMPLE 9 Prove that the medians of an equilateral triangle are equal. GIVEN A AABC in which AB = BC = AC and AD, BE and CF are its medians.​

Answer 1

Answer:

Consider an equilateral △ABC,

Let D, E, F are midpoints of BC, CA and AB. Here, AD, BE and CF are medians of △ABC. Now, D is midpoint of BC => BD = DC Similarly, CE = EA and AF = FB Since ΔABC is an equilateral triangle AB = BC = CA …..

(i) BD = DC = CE = EA = AF = FB …………

(ii) And also, ∠ ABC = ∠ BCA = ∠ CAB = 60° ……….(iii) Consider Δ ABD and Δ BCE AB = BC [From (i)] BD = CE [From (ii)] ∠ ABD = ∠ BCE [From (iii)] By SAS congruence criterion, Δ ABD ≃ Δ BCE => AD = BE ……..

(iv) [Corresponding parts of congruent triangles are equal in measure] Now, consider Δ BCE and Δ CAF, BC = CA [From (i)] ∠ BCE = ∠ CAF [From (ii)] CE = AF [From (ii)] By SAS congruence criterion, Δ BCE ≃ Δ CAF BE = CF …………..

(v) [Corresponding parts of congruent triangles are equal] From (iv) and (v), we have AD = BE = CF Median AD = Median BE = Median CF The medians of an equilateral triangle are equal. Hence proved.