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The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB?
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The figure, there are two points D and E on side AB of triangle ABC su...
Given Information
- Triangle ABC with points D and E on side AB such that AD = BE.
- Line DP is parallel to BC, and line EQ is parallel to AC.
Objective
- To prove that line PQ is parallel to side AB.
Proof Steps
1. Parallel Lines and Proportional Segments
- Since DP || BC and EQ || AC, by the Basic Proportionality Theorem (Thales' theorem), we can establish that:
- AD/DB = AP/PC (from DP || BC)
- BE/EA = EQ/QA (from EQ || AC)
2. Using the Property of Similar Triangles
- By the properties of parallel lines, triangles ABD and ABE are similar to triangles APD and EQB respectively.
- Thus, the angles formed create equal ratios.
3. Establishing the Ratios
- From the similarity of triangles:
- AD/BE = AP/PQ (from triangle ABD and triangle APQ)
- Since AD = BE, we have:
- AP/PQ = 1 (indicating AP = PQ)
4. Conclusion: Proving PQ || AB
- Since the segments AP and PQ are equal and the triangles are similar, we conclude that PQ is parallel to AB.
- Therefore, PQ || AB, as required.
Final Result
- Thus, we have proven that line PQ is parallel to line AB in triangle ABC, given the conditions of parallel lines and equal segments along side AB.
- Triangle ABC with points D and E on side AB such that AD = BE.
- Line DP is parallel to BC, and line EQ is parallel to AC.
Objective
- To prove that line PQ is parallel to side AB.
Proof Steps
1. Parallel Lines and Proportional Segments
- Since DP || BC and EQ || AC, by the Basic Proportionality Theorem (Thales' theorem), we can establish that:
- AD/DB = AP/PC (from DP || BC)
- BE/EA = EQ/QA (from EQ || AC)
2. Using the Property of Similar Triangles
- By the properties of parallel lines, triangles ABD and ABE are similar to triangles APD and EQB respectively.
- Thus, the angles formed create equal ratios.
3. Establishing the Ratios
- From the similarity of triangles:
- AD/BE = AP/PQ (from triangle ABD and triangle APQ)
- Since AD = BE, we have:
- AP/PQ = 1 (indicating AP = PQ)
4. Conclusion: Proving PQ || AB
- Since the segments AP and PQ are equal and the triangles are similar, we conclude that PQ is parallel to AB.
- Therefore, PQ || AB, as required.
Final Result
- Thus, we have proven that line PQ is parallel to line AB in triangle ABC, given the conditions of parallel lines and equal segments along side AB.
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The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB?
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The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB?.
The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The figure, there are two points D and E on side AB of triangle ABC such that AD = BE If DP ||BC and EQ ||AC then prove that PQ ||AB?.
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