Question

ABC is an isosceles triangle with AB= AC ,D ,E ,F are midpoints of BC,CA,AB respectively. Show that AD is prependicular to EF and AD is bisected by EF .

Answer 1

use this photo .hope I helped you.

Answer 2

Given :-

ΔABC is isosceles triangle with AB = AC.

D, E and F are the mid points of BC, CA and AB respectively.

To be prove :

AD ⊥ EF and bisected by it.

Construction : Join D, E and F

PROOF :

DE║AB and DE = ¹/₂ AB

and

DF║AC and DF = ¹/₂ AC

[ ∵ Line segment joining midpoints of  two sides of a triangle is parallel to the third side and half of it ]

DE = DF       [ ∴ Ab = AC ]

And

AF = AE       [ ∴ AF = ¹/₂ AB, AE = ¹/₂ AC ]

∴ DE = AE = AF = DF and also DF║AE and DE║AF

⇒ DEAF is a rhombus

Since, diagonals of a rhombus bisect each other at right angles.

∴ AD ⊥ EF and is bisected by it.