Question

In the given fig.,BE perpendicular AC. AD is any line from A to BC intersecting BE at H. P, Q and R are mid-points of AH, AB and BC respectively, then prove that angle PQR=90o.

Answer 1

Hi!

If your question is “In a triangle ABC, BE is perpendicular to AC and AD is any line from A to BC intersecting BE at H. If P, Q and R are mid points of AH, AB and BC respectively then prove that angle PQR = 90 degree.”, then the answer to this question is as follows:

Let ABC be a triangle.
We have BE⊥AC and AD is any line from A intersecting AE at H, P, Q and R are the mid points of AH, AB and BC respectively.

Given Q and R are the mid points of sides AB and BC of ∆ABC respectively
∴ QR||AC [Mid-point theorem]
Given BE⊥AC
∴ BE⊥QR
In ∆ABH and Q and P are the mid points of sides AB and AH respectively.
∴ QP||BH [Mid-point theorem]
∴QP||BE
⇒QP⊥QR (BE⊥QR)
⇒∠PQR = 90°


hopes this helps!!!!!!!

Answer 2

In  ΔABC,  Q & R are the mid-points of AB and BC respectively.

∴  QR||AC -------- (i)

In  ΔABH , Q and P are the mid-points of AB and AH respectively.

∴  QP||BH  ⇒  QP||BE ------- (ii)

But AC ⊥ BE . Therefore, from (i) & (ii), we have

QP⊥QR ⇒∠PQR=90°

Hope this helps