In figure, BE⊥ AC, AD is any line from A to BC intersecting BE in H. P, Q
Question:

In figure, BE⊥ AC, AD is any line from A to BC intersecting BE in H. P, Q and R are respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°

Solution:

Given,

BE ⊥ AC and P, Q and R are respectively mid-point of AH, AB and BC.

To prove: ∠PQR = 90°

Proof: In ΔABC, Q and R are 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   ….. (ii)

But, BE⊥AC

Therefore, from equation (i) and equation (ii) we have,

QP⊥QR

⟹ ∠PQR = 90°

Hence Proved.