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प्रश्न
Prove that the medians of an equilateral triangle are equal.
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उत्तर
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
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