▪️From a point P outside a circle with centre O, tangents PA and PB are drawn to the circle. Prove that OP is the right bisector of the line segment AB.
Answers
Qᴜᴇꜱᴛɪᴏɴ :
➥ From a point P outside a circle with centre O, tangents PA and PB are drawn to the circle. Prove that OP is the right bisector of the line segment AB.
Pʀᴏᴠᴇᴅ :
➥ OP is the right bisector of AB
Gɪᴠᴇɴ :
➤ PA and PB tangents to a circle with centre O from an external point P
Tᴏ Pʀᴏᴠᴇ :
➤ OP is the right bisector of AB ?
Cᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ :
➤ Join AB. Let AB intersect OP at M.
Pʀᴏᴏꜰ :
In ∆MAP and ∆MPB, we have
⪼ PA = PB [ ∵ tangents to a circle from an external point are equal ]
⪼ MP = MP [ Common ]
⪼ ∠MPA = ∠MPB [ ∵ tangents from an external point are equally inclined to the line segment joining the center to that point , i.e., ∠OPA = ∠OPB ]
∴ ∆MAP ≅ ∆MPB [ By SAS-congruence ]
And so, MA = MB [ CPCT ]
and ∠AMP = ∠BMP [ CPCT ]
But, ∠AMP + ∠BMP = 180° [ linear pair ]
∴ ∠AMP = ∠BMP = 90°
Hence, OP is the right bisector of AB.
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Additional Information :
What is the congruence of traingle ?
➜ Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.
What is the Full Form of CPCT ?
➜ CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.
What are the Rules of Congruency ?
➜ There are 5 main rules of congruency for triangles:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- RHS ( Right angled-Hypotenuse-Side)
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Answer:
Qᴜᴇꜱᴛɪᴏɴ :
➥ From a point P outside a circle with centre O, tangents PA and PB are drawn to the circle. Prove that OP is the right bisector of the line segment AB.
Pʀᴏᴠᴇᴅ :
➥ OP is the right bisector of AB
Gɪᴠᴇɴ :
➤ PA and PB tangents to a circle with centre O from an external point P
Tᴏ Pʀᴏᴠᴇ :
➤ OP is the right bisector of AB ?
Cᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ :
➤ Join AB. Let AB intersect OP at M.
Pʀᴏᴏꜰ :
In ∆MAP and ∆MPB, we have
⪼ PA = PB [ ∵ tangents to a circle from an external point are equal ]
⪼ MP = MP [ Common ]
⪼ ∠MPA = ∠MPB [ ∵ tangents from an external point are equally inclined to the line segment joining the center to that point , i.e., ∠OPA = ∠OPB ]
∴ ∆MAP ≅ ∆MPB [ By SAS-congruence ]
And so, MA = MB [ CPCT ]
and ∠AMP = ∠BMP [ CPCT ]
But, ∠AMP + ∠BMP = 180° [ linear pair ]
∴ ∠AMP = ∠BMP = 90°
Hence, OP is the right bisector of AB.
⠀
▭▬▭▬▭▬▭▬▭▬▭▬▭▬▭▬▭▬
Additional Information :
What is the congruence of traingle ?
➜ Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.
What is the Full Form of CPCT ?
➜ CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.
What are the Rules of Congruency ?
➜ There are 5 main rules of congruency for triangles:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- RHS ( Right angled-Hypotenuse-Side)
▭▬▭▬▭▬▭▬▭▬▭▬▭▬▭▬▭▬
Step-by-step explanation: