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Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD?
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Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect e...
Understanding the Trapezium and Its Properties
In trapezium ABCD, where AB is parallel to DC, the diagonals AC and BD intersect at point O. To prove that \( \frac{OA}{OC} = \frac{OB}{OD} \), we can use the similarity criterion for triangles.
Triangles Involved
- Triangles \( \triangle AOB \) and \( \triangle COD \) share a common angle at point O.
- The angles \( \angle AOB \) and \( \angle COD \) are equal, as they are alternate interior angles formed by the transversal lines AB and DC.
Using the Parallel Lines Property
- Since AB is parallel to DC, we have:
- \( \angle OAB = \angle OCD \) (corresponding angles)
- \( \angle OBA = \angle ODC \) (corresponding angles)
This establishes that triangles \( \triangle AOB \) and \( \triangle COD \) are similar by the AA (Angle-Angle) criterion.
Ratios from Similar Triangles
- From the similarity of triangles, we can write:
\[
\frac{OA}{OC} = \frac{OB}{OD}
\]
This results from the property that the ratios of corresponding sides in similar triangles are equal.
Conclusion
- The relationship \( \frac{OA}{OC} = \frac{OB}{OD} \) is a direct consequence of the properties of parallel lines and the similarity of triangles formed by the intersecting diagonals.
- This proof demonstrates the fundamental geometric principles that govern trapeziums and their diagonals, providing a clear understanding of the relationships within the figure.
In trapezium ABCD, where AB is parallel to DC, the diagonals AC and BD intersect at point O. To prove that \( \frac{OA}{OC} = \frac{OB}{OD} \), we can use the similarity criterion for triangles.
Triangles Involved
- Triangles \( \triangle AOB \) and \( \triangle COD \) share a common angle at point O.
- The angles \( \angle AOB \) and \( \angle COD \) are equal, as they are alternate interior angles formed by the transversal lines AB and DC.
Using the Parallel Lines Property
- Since AB is parallel to DC, we have:
- \( \angle OAB = \angle OCD \) (corresponding angles)
- \( \angle OBA = \angle ODC \) (corresponding angles)
This establishes that triangles \( \triangle AOB \) and \( \triangle COD \) are similar by the AA (Angle-Angle) criterion.
Ratios from Similar Triangles
- From the similarity of triangles, we can write:
\[
\frac{OA}{OC} = \frac{OB}{OD}
\]
This results from the property that the ratios of corresponding sides in similar triangles are equal.
Conclusion
- The relationship \( \frac{OA}{OC} = \frac{OB}{OD} \) is a direct consequence of the properties of parallel lines and the similarity of triangles formed by the intersecting diagonals.
- This proof demonstrates the fundamental geometric principles that govern trapeziums and their diagonals, providing a clear understanding of the relationships within the figure.
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Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD?
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Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD?.
Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Diagonal AC and BD of a trapezium ABCD with AB parallel DC intersect each other at the point O using a similarly criterion for two triangles show that OA/OC=OB/OD?.
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