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.
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