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In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.?
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In a triangle ABC, BM and CN are perpendicular from B and C respective...
Understanding the Triangle Configuration
In triangle ABC, lines BM and CN are drawn perpendicular to a line passing through point A. Here, M and N are the feet of the perpendiculars from points B and C respectively. L is defined as the midpoint of side BC.
Key Points to Consider
- Midpoint Definition: L is the midpoint of segment BC, which means that BL = LC.
- Perpendicular Distances: Since BM and CN are perpendicular to the same line, the distances from points M and N to the line through A can be visualized as vertical distances.
Proving ML = NL
1. Identifying Midpoint Relation:
- Since L is the midpoint, we have BL = LC.
2. Distance Relation:
- The perpendiculars from B and C to the line through A create right triangles (ABM and ACN) where BM and CN are altitudes.
3. Using Right Triangles:
- The distances ML and NL can be calculated using the relationships in these right triangles. Since BM and CN are perpendicular, triangles ABM and ACN share the same height from point A to line BC.
4. Conclusion:
- Since both ML and NL are defined as the distances from the midpoint L to the lines BM and CN, and these lines are symmetric about L, we can conclude that ML = NL.
Final Result
This symmetry in the configuration leads to the conclusion that ML equals NL, demonstrating that the distances from the midpoint L to the perpendiculars from B and C are indeed equal. Hence, ML = NL is proven.
In triangle ABC, lines BM and CN are drawn perpendicular to a line passing through point A. Here, M and N are the feet of the perpendiculars from points B and C respectively. L is defined as the midpoint of side BC.
Key Points to Consider
- Midpoint Definition: L is the midpoint of segment BC, which means that BL = LC.
- Perpendicular Distances: Since BM and CN are perpendicular to the same line, the distances from points M and N to the line through A can be visualized as vertical distances.
Proving ML = NL
1. Identifying Midpoint Relation:
- Since L is the midpoint, we have BL = LC.
2. Distance Relation:
- The perpendiculars from B and C to the line through A create right triangles (ABM and ACN) where BM and CN are altitudes.
3. Using Right Triangles:
- The distances ML and NL can be calculated using the relationships in these right triangles. Since BM and CN are perpendicular, triangles ABM and ACN share the same height from point A to line BC.
4. Conclusion:
- Since both ML and NL are defined as the distances from the midpoint L to the lines BM and CN, and these lines are symmetric about L, we can conclude that ML = NL.
Final Result
This symmetry in the configuration leads to the conclusion that ML equals NL, demonstrating that the distances from the midpoint L to the perpendiculars from B and C are indeed equal. Hence, ML = NL is proven.
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In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.?
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In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.?.
In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC, BM and CN are perpendicular from B and C respectively on any line passing through A. If L is the mid point of BC, prove that ML=NL.?.
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