Question

Prove that the medians of an equilateral triangle are equal. 

Answer 1

Given to prove that the medians of an equilateral triangle are equal
Median: The line joining the vertex and midpoint of opposite side.
Now, consider an equilateral triangle ABC
Let D,E,F are midpoints of , BC CAand . AB
Then, , AD BE and CF are medians of . Δ ABC 

Now , 

D is midpoint of BC⇒ BD = DC =`(BC)/2` 

Similarly ,` CE=EA=(AC)/2` 

`AF= FB=(AB)/2` 

Since Δ ABC is an equilateral traingle ⇒ AB=BC= CA      .............(1) 

`BD=DC=CE=EA=AF=FB= (BC)/2=(AC)/2 (AB)/2` ..............(2) 

And also , ∠ ABC= ∠ BCA=∠CAB=60°    ..................(3) 

Now, consider Δ ABD and Δ BCE 

AB=BC                  [from (1)] 

BD= CE                 [from (2)] 

∠ ABD= ∠  BCE            [from (3)] [∠ ABD and ∠ ABC and ∠ BCE and BCA aare same ] 

So, from SAS congruence criterion , we  have 

Δ ABD ≅ Δ BCE 

AD= BE                          ........................(4) 

[corresponding parts of congruent triangles are equal] 

Now, consider ΔBCE and Δ CAF, 

BC = CA                           [from (1)]

∠BCE =∠ CAF                  [from (3)]

 [∠  BCE and  ∠ BCA and ∠ CAF annd ∠ CAB are same ] 

CE=AF                      [from (2)] 

So, from SAS congruence criterion, we have Δ BCE≅ Δ CAF 

⇒ BE=CF                   ..........................(5) 

[Corresponding parts of congruent triangles are equal ] 

From (4) and (5), we have 

AD =BE= CF
⇒Median AD = Median BE = Median CF
∴The medians of an equilateral triangle are equal
∴Hence proved