Question

Prove that tangent drawn at the end points of a chord of a circle make equal angles with the chord

Answer 1

Answer:

Step-by-step explanation:

First prove ΔPAO ≅ PBO

1. From ΔPAO,ΔPBO
2. AO = BO (radius)
3. ∠OAP = ∠OBP = 90° (radii make right angle with the tangents )
4. PO =PO (common side )
5. As per Side Side Side Congruence ΔPAO ≅ ΔPBO
6. Corresponding parts of congruent triangles are equal ∴∠APO = ∠BPO

Now let the line segment PO intersects chord AB at E

1. Now from ΔAEP , ΔBEP
2. ∠AEP =∠BEP =90° (radius bisects the chord perpendicularly
3. ∠APE = ∠BPE (proved above )
4. As per Angle Angle Similarity ΔAEP is similar to ΔBEP
5. So, from the above ∠EAP = ∠EBP
6. ∠EAP,∠EBP are the angles made by the tangents with the chord
7. Hence the tangent drawn at the end of a chord makes equal angles with the chord