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Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.?
Most Upvoted Answer
Points P, Q and R lie on sides BC, CA and AB respectively of triangle ...
Proof:
Given: Triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
To prove: RP||CA.
Proof:
Step 1: Draw a diagram of the given triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
Step 2: Draw a line segment from point R to point C.
Step 3: Since PQ||AB and QR||BC, we have ∠QRC = ∠ABC and ∠PQR = ∠BAC (corresponding angles).
Step 4: Also, since RP is a transversal cutting parallel lines BC and PQ, we have ∠RPA = ∠PQC (alternate angles).
Step 5: Adding ∠QRC and ∠RPA, we get:
∠QRC + ∠RPA = ∠ABC + ∠PQC (using corresponding angles)
Step 6: Simplifying the above equation, we get:
∠QRP = ∠ACB
Step 7: Since ∠QRP and ∠ACB are alternate interior angles, we have RP||CA.
Step 8: Therefore, RP||CA is proved.
Conclusion: Hence, it is proved that RP||CA in the given triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
Given: Triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
To prove: RP||CA.
Proof:
Step 1: Draw a diagram of the given triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
Step 2: Draw a line segment from point R to point C.
Step 3: Since PQ||AB and QR||BC, we have ∠QRC = ∠ABC and ∠PQR = ∠BAC (corresponding angles).
Step 4: Also, since RP is a transversal cutting parallel lines BC and PQ, we have ∠RPA = ∠PQC (alternate angles).
Step 5: Adding ∠QRC and ∠RPA, we get:
∠QRC + ∠RPA = ∠ABC + ∠PQC (using corresponding angles)
Step 6: Simplifying the above equation, we get:
∠QRP = ∠ACB
Step 7: Since ∠QRP and ∠ACB are alternate interior angles, we have RP||CA.
Step 8: Therefore, RP||CA is proved.
Conclusion: Hence, it is proved that RP||CA in the given triangle ABC with points P, Q and R on sides BC, CA and AB respectively such that PQ||AB and QR||BC.
Community Answer
Points P, Q and R lie on sides BC, CA and AB respectively of triangle ...
In triangle ABC PQ||AB =)AQ/QC=CP/PB----(1)(BPT) now QR||BC=) AR/RB=AQ/QC-----(2)(BPT) from(1) and(2) we get AR/RB=CP/PB=) RP||CA(converse of BPT)[proved]
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Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.?
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Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.?.
Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Points P, Q and R lie on sides BC, CA and AB respectively of triangle ABC such that PQ||AB and QR||BC, prove that RP||CA.?.
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