Property of Median Drawn on Hypotenuse of the Right Triangle
Prove that th...
Question
Prove that the medians of an equilateral triangle are equal.
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Solution
Let the ΔABC be equilateral triangle with medians AE,FB,CD.
Consider the triangles ΔABF and ΔBAE.
Since AC=BC ⇒AC2=BC2 [Dividing 2 on both sides] Thus, AF=BE ∠FAB=∠EBA [ Since, in equilateral triangle, all three are equal] AB=AB (common side) Thus, by SAS criterion, we get ΔABF≅ΔBAE Therefore, AE=BF ........(i) Similar way, we can prove CD=AE..........(ii), by taking AB=BC From equations(i) and (ii), we get AE=BF=CD Hence, the the medians of equilateral triangle are equal.