Question

BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If L is the mid-point of BC, prove that LM = LN.

Answer 1

Draw LS perpendicular to line MN,

∴ The line BM,LS,CN being the same perpendiculars,on line MN are parallel to each other

We have, 

BL = LC (L is mid point of BC) 

Using intercept theorem,

MS = SN   …(i)

Now, 

In ∆MLS and ∆LSN

MS = SN

∠LSM =∠LSN = 90° 

(LS perpendicular to MN) 

And, 

SL = LS is common

∴∆MLS ≅∆LSN 

(SAS congruency)

∴ LM = LN 

Proved.