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In the fig. X is the mid point of the side RS of a parallelogram PQRS. A line through R parallel to PX intersect PQ at Y and SP produce at Z . Show that PS = PZ and RY = YZ.?
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In the fig. X is the mid point of the side RS of a parallelogram PQRS....
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In the fig. X is the mid point of the side RS of a parallelogram PQRS....
Given:
- PQRS is a parallelogram.
- X is the midpoint of RS.
- A line through R parallel to PX intersects PQ at Y.
- The line through R parallel to PX intersects SP produced at Z.
To prove:
- PS = PZ
- RY = YZ
Proof:
Step 1: Extend QY to meet SR at M.
Step 2: Prove that QM || PX
- Since PQRS is a parallelogram, PQ || SR.
- By alternate interior angles, ∠QRY = ∠PXS.
- Also, ∠RYQ = ∠XSP (corresponding angles).
- Therefore, ∠QRY = ∠PXS and ∠RYQ = ∠XSP.
- By AA similarity, ΔQRY ~ ΔPXS.
- Therefore, ∠QYR = ∠PXS.
- Since ∠QYR and ∠PXS are corresponding angles, QM || PX.
Step 3: Prove that QZ || PX
- Since QM || PX, and Z lies on QM, Z must also be parallel to PX.
Step 4: Prove that PZ = PS
- Since QZ || PX and PQ || SR, by the converse of the alternate interior angles theorem, QZ || SR.
- Therefore, QZRS is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, PZ = SR.
- Since PQRS is a parallelogram, PQ = SR.
- Therefore, PZ = PQ.
- Also, PZ || QY, so ∠PZQ = ∠QYR.
- Similarly, PQ || RY, so ∠PQZ = ∠RYQ.
- By AA similarity, ΔPZQ ~ ΔRYQ.
- Therefore, ∠ZPQ = ∠YQR.
- Since ∠ZPQ and ∠YQR are corresponding angles, PZ || RY.
- Therefore, PZRY is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, PZ = RY.
Step 5: Prove that RY = YZ
- Since PZRY is a parallelogram, RY = PZ.
- Since QZRS is a parallelogram, PZ = QS.
- Therefore, RY = QS.
- Since QY || SR, by the alternate interior angles theorem, ∠RYQ = ∠RQS.
- Therefore, RYQ is an isosceles triangle.
- In an isosceles triangle, the base angles are equal.
- Therefore, ∠YQR = ∠YRQ.
- Since ∠YQR and ∠YRQ are corresponding angles, YQ || SR.
- By the converse of the alternate interior angles theorem, QY || SR.
- Therefore, QY || SR, and QYRS is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, RY = QS = YZ.
Conclusion:
- From the above proof
- PQRS is a parallelogram.
- X is the midpoint of RS.
- A line through R parallel to PX intersects PQ at Y.
- The line through R parallel to PX intersects SP produced at Z.
To prove:
- PS = PZ
- RY = YZ
Proof:
Step 1: Extend QY to meet SR at M.
Step 2: Prove that QM || PX
- Since PQRS is a parallelogram, PQ || SR.
- By alternate interior angles, ∠QRY = ∠PXS.
- Also, ∠RYQ = ∠XSP (corresponding angles).
- Therefore, ∠QRY = ∠PXS and ∠RYQ = ∠XSP.
- By AA similarity, ΔQRY ~ ΔPXS.
- Therefore, ∠QYR = ∠PXS.
- Since ∠QYR and ∠PXS are corresponding angles, QM || PX.
Step 3: Prove that QZ || PX
- Since QM || PX, and Z lies on QM, Z must also be parallel to PX.
Step 4: Prove that PZ = PS
- Since QZ || PX and PQ || SR, by the converse of the alternate interior angles theorem, QZ || SR.
- Therefore, QZRS is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, PZ = SR.
- Since PQRS is a parallelogram, PQ = SR.
- Therefore, PZ = PQ.
- Also, PZ || QY, so ∠PZQ = ∠QYR.
- Similarly, PQ || RY, so ∠PQZ = ∠RYQ.
- By AA similarity, ΔPZQ ~ ΔRYQ.
- Therefore, ∠ZPQ = ∠YQR.
- Since ∠ZPQ and ∠YQR are corresponding angles, PZ || RY.
- Therefore, PZRY is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, PZ = RY.
Step 5: Prove that RY = YZ
- Since PZRY is a parallelogram, RY = PZ.
- Since QZRS is a parallelogram, PZ = QS.
- Therefore, RY = QS.
- Since QY || SR, by the alternate interior angles theorem, ∠RYQ = ∠RQS.
- Therefore, RYQ is an isosceles triangle.
- In an isosceles triangle, the base angles are equal.
- Therefore, ∠YQR = ∠YRQ.
- Since ∠YQR and ∠YRQ are corresponding angles, YQ || SR.
- By the converse of the alternate interior angles theorem, QY || SR.
- Therefore, QY || SR, and QYRS is a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, RY = QS = YZ.
Conclusion:
- From the above proof
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In the fig. X is the mid point of the side RS of a parallelogram PQRS. A line through R parallel to PX intersect PQ at Y and SP produce at Z . Show that PS = PZ and RY = YZ.?
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In the fig. X is the mid point of the side RS of a parallelogram PQRS. A line through R parallel to PX intersect PQ at Y and SP produce at Z . Show that PS = PZ and RY = YZ.? 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 the fig. X is the mid point of the side RS of a parallelogram PQRS. A line through R parallel to PX intersect PQ at Y and SP produce at Z . Show that PS = PZ and RY = YZ.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the fig. X is the mid point of the side RS of a parallelogram PQRS. A line through R parallel to PX intersect PQ at Y and SP produce at Z . Show that PS = PZ and RY = YZ.?.
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