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Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO?
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Abcd is a trapezium in which AB is parallel to DC and it's diagonals i...
Given information:
- ABCD is a trapezium.
- AB is parallel to DC.
- The diagonals of ABCD intersect at point O.
To prove:
AO/BO = CO/DO
Proof:
Step 1: Triangles AOB and COD are similar.
- In ΔAOB and ΔCOD, angle AOB = angle COD (vertically opposite angles).
- AB is parallel to DC (given).
- Therefore, angle ABO = angle CDO (alternate interior angles).
- Hence, ΔAOB ~ ΔCOD (by AA similarity criterion).
Step 2: Using the property of similar triangles.
- In similar triangles, the ratio of corresponding sides is equal.
- Therefore, we have AO/CO = BO/DO (corresponding sides of similar triangles AOB and COD).
Step 3: Rearranging the equation.
- Cross multiplying the above equation, we get AO * DO = BO * CO.
Step 4: Dividing by AO * BO.
- Dividing both sides of the equation by AO * BO, we get (AO * DO) / (AO * BO) = (BO * CO) / (AO * BO).
- Simplifying, we get DO/BO = CO/AO.
Step 5: Cross multiplying.
- Cross multiplying the above equation, we get DO * AO = BO * CO.
Step 6: Dividing by CO * DO.
- Dividing both sides of the equation by CO * DO, we get (DO * AO) / (CO * DO) = (BO * CO) / (CO * DO).
- Simplifying, we get AO/CO = BO/DO.
Conclusion:
- From the above steps, we have proved that AO/BO = CO/DO.
- ABCD is a trapezium.
- AB is parallel to DC.
- The diagonals of ABCD intersect at point O.
To prove:
AO/BO = CO/DO
Proof:
Step 1: Triangles AOB and COD are similar.
- In ΔAOB and ΔCOD, angle AOB = angle COD (vertically opposite angles).
- AB is parallel to DC (given).
- Therefore, angle ABO = angle CDO (alternate interior angles).
- Hence, ΔAOB ~ ΔCOD (by AA similarity criterion).
Step 2: Using the property of similar triangles.
- In similar triangles, the ratio of corresponding sides is equal.
- Therefore, we have AO/CO = BO/DO (corresponding sides of similar triangles AOB and COD).
Step 3: Rearranging the equation.
- Cross multiplying the above equation, we get AO * DO = BO * CO.
Step 4: Dividing by AO * BO.
- Dividing both sides of the equation by AO * BO, we get (AO * DO) / (AO * BO) = (BO * CO) / (AO * BO).
- Simplifying, we get DO/BO = CO/AO.
Step 5: Cross multiplying.
- Cross multiplying the above equation, we get DO * AO = BO * CO.
Step 6: Dividing by CO * DO.
- Dividing both sides of the equation by CO * DO, we get (DO * AO) / (CO * DO) = (BO * CO) / (CO * DO).
- Simplifying, we get AO/CO = BO/DO.
Conclusion:
- From the above steps, we have proved that AO/BO = CO/DO.
Community Answer
Abcd is a trapezium in which AB is parallel to DC and it's diagonals i...
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Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO?
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Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO? 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 Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO?.
Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO? 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 Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Abcd is a trapezium in which AB is parallel to DC and it's diagonals interest each other at the point O . Show that AO/BO =CO/DO?.
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