In this chapter we will prove that median of equilateral triangle are equal in length.
Given below is an equilateral triangle in which AB = BC= CA
Lines AD, BE & CF are the medians of the triangle.
Median divide the side into equal halves.
AF = FB
BD = DC
AE = EC
To prove:
All the medians are of equal length.
AD = BE = CF
Solution
Since all sides are equal, the half length of given sides are also equal.
AF = FB = BD = DC = CE = EA
Now consider triangle ABD and BCE;
AB = BC
∠B = ∠C = 60 degree ( equilateral triangle)
BD = CE
By SAS congruency, both the triangles are congruent.
Hence AD = BE
Now consider triangle BCE and CAF
BC = CA
∠C = ∠A
CE = AF
By SAS congruency, both the triangles are congruent.
Hence, BE = CF
Combining all the results we get;
AD = BE = CF
Hence inn equilateral triangle, all the medians are of equal length.