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The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD?
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The diagonals AC and BD of a parallelogram ABCD intersect at O . If P ...
Proof of PO Parallel to AB
To establish that PO is parallel to AB, consider the properties of parallelograms:
- Diagonals of a parallelogram bisect each other. Thus, O is the midpoint of both diagonals AC and BD.
- Since P is the midpoint of AD, we can use the midpoint theorem.
Using the properties:
- Triangle AOD and triangle BOC are congruent (by the Side-Angle-Side postulate).
- Therefore, AO = OC and BO = OD.
This implies that line segments OP and AB are corresponding segments in similar triangles, leading to:
- PO is parallel to AB.
Proof that PO = 1/2 CD
Next, we need to show that PO equals half the length of CD:
- Since O is the midpoint of both diagonals, we know that AO = OC and BO = OD.
- By the midpoint theorem, if P is the midpoint of AD, then:
- AP = PD
- Since AB is parallel to CD, the length of PO can be calculated using the property of midpoints in triangles.
Using the triangle similarity:
- PO = 1/2 (CD), as the midpoints divide the line segments proportionally.
Thus, combining these proofs, we conclude that:
- PO is parallel to AB.
- PO equals 1/2 the length of CD.
This completes the proof.
To establish that PO is parallel to AB, consider the properties of parallelograms:
- Diagonals of a parallelogram bisect each other. Thus, O is the midpoint of both diagonals AC and BD.
- Since P is the midpoint of AD, we can use the midpoint theorem.
Using the properties:
- Triangle AOD and triangle BOC are congruent (by the Side-Angle-Side postulate).
- Therefore, AO = OC and BO = OD.
This implies that line segments OP and AB are corresponding segments in similar triangles, leading to:
- PO is parallel to AB.
Proof that PO = 1/2 CD
Next, we need to show that PO equals half the length of CD:
- Since O is the midpoint of both diagonals, we know that AO = OC and BO = OD.
- By the midpoint theorem, if P is the midpoint of AD, then:
- AP = PD
- Since AB is parallel to CD, the length of PO can be calculated using the property of midpoints in triangles.
Using the triangle similarity:
- PO = 1/2 (CD), as the midpoints divide the line segments proportionally.
Thus, combining these proofs, we conclude that:
- PO is parallel to AB.
- PO equals 1/2 the length of CD.
This completes the proof.
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The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD?
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The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD? 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 The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD?.
The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD? 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 The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The diagonals AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD, prove that (1) PO parallel AB (2) PO =1/2CD?.
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