Question

prove if tangents diagonals of a cyclic quadrilateral are perpendicular to each other show that the line passing through the point of intersection of diagonals and midpoint of a side is perpendicular to the opposite side .​

Answer 1

1

Given: A cyclic quadrilateral ABCD in which Diagonals AB and CD intersect each other at point O. Also, AP=BP, and ∠APO=∠BPO=90°

To prove: PQ ⊥CD.

Proof: In ΔAPO and ΔBPO

∠APO=∠BPO=90°→→(given)

AP=BP→→Given

Side PO is common.

ΔAPO ≅ ΔBPO→→[SAS]

∠AOP=∠BOP→→[CPCT]

∠AOB=90°

2∠AOP=90°

∠AOP=45°

In Δ APO

∠OAP +∠APO+∠POA=180°→→Angle sum property of triangle

∠OAP+90°+45°=180°

∠OAP=45°

Similarly, ∠OBP=45°

∠AOP=∠COQ=45°→Vertically opposite angles

∠BOP=∠DOQ=45°→Vertically opposite angles

Also, angle in the same segment of a circle are equal.

∠BDC=∠CAB=45°

∠ABD=∠ACD=45°

Now, in ΔOQD and ΔOQC

45°+∠OQD +45°=180°→→→Angle sum property of triangle

∠OQD=180°-90°

∠OQD=90°

Similarly, ∠OQC=90°

So, we can conclude that , PQ⊥CD.