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ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle
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ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, B...
Given- ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.
To Prove-PQRS is a rectangle
Construction,
AC and BD are joined.
Proof,
In ΔDRS and ΔBPQ,
DS = BQ (Halves of the opposite sides of the rhombus)
∠SDR = ∠QBP (Opposite angles of the rhombus)
DR = BP (Halves of the opposite sides of the rhombus)
Thus, ΔDRS ≅ ΔBPQ by SAS congruence condition.
RS = PQ by CPCT -- (i)
In ΔQCR and ΔSAP,
RC = PA (Halves of the opposite sides of the rhombus)
∠RCQ = ∠PAS (Opposite angles of the rhombus)
CQ = AS (Halves of the opposite sides of the rhombus)
Thus, ΔQCR ≅ ΔSAP by SAS congruence condition.
RQ = SP by CPCT -- (ii)
Now,
In ΔCDB,
R and Q are the mid points of CD and BC respectively.
⇒ QR || BD
also,
P and S are the mid points of AD and AB respectively.
⇒ PS || BD
⇒ QR || PS
Thus, PQRS is a parallelogram.
also, ∠PQR = 90degree
Now,
In PQRS,
RS = PQ and RQ = SP from (i) and (ii)
∠Q = 90degree
Thus, PQRS is a rectangle.
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, B...
Proof:
Given: ABCD is a rhombus with P, Q, R, and S as the midpoints of AB, BC, CD, and DA respectively.
To prove: PQRS is a rectangle.
Proof:
1. Properties of a Rhombus:
- A rhombus is a quadrilateral with all sides of equal length.
- Opposite sides of a rhombus are parallel.
- Diagonals of a rhombus bisect each other at right angles.
2. Properties of Midpoints:
- The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
3. Proving PQRS is a Rectangle:
- Let's consider triangle ABC. Since P is the midpoint of AB, the segment PS joins the midpoints of sides AB and AD. According to the properties of midpoints, PS is parallel to side BD and half its length.
- Similarly, the segment QR joins the midpoints of sides BC and CD. According to the properties of midpoints, QR is parallel to side BD and half its length.
- Therefore, PS and QR are parallel and have equal lengths. This means that PQRS is a parallelogram.
4. Proving PQRS is a Rectangle (continued):
- Next, let's consider triangle ABD. Since R is the midpoint of CD, the segment QR joins the midpoints of sides CD and DA. According to the properties of midpoints, QR is parallel to side AB and half its length.
- Similarly, the segment PS joins the midpoints of sides AB and BC. According to the properties of midpoints, PS is parallel to side AB and half its length.
- Therefore, QR and PS are parallel and have equal lengths. This means that PQRS is a parallelogram.
5. Proving PQRS is a Rectangle (continued):
- From the previous steps, we have established that PQRS is a parallelogram.
- Now, let's consider the diagonals of PQRS. Diagonals PR and QS bisect each other at point O.
- Since PQRS is a parallelogram, its diagonals bisect each other.
- Moreover, since PR and QS are parallel, they are also perpendicular to each other.
- Therefore, PQRS is a parallelogram with perpendicular diagonals, which satisfies the definition of a rectangle.
Conclusion:
- We have proved that PQRS is a rectangle using the properties of a rhombus, the properties of midpoints, and the properties of a parallelogram with perpendicular diagonals.
- Thus, the claim is proven.
Given: ABCD is a rhombus with P, Q, R, and S as the midpoints of AB, BC, CD, and DA respectively.
To prove: PQRS is a rectangle.
Proof:
1. Properties of a Rhombus:
- A rhombus is a quadrilateral with all sides of equal length.
- Opposite sides of a rhombus are parallel.
- Diagonals of a rhombus bisect each other at right angles.
2. Properties of Midpoints:
- The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
3. Proving PQRS is a Rectangle:
- Let's consider triangle ABC. Since P is the midpoint of AB, the segment PS joins the midpoints of sides AB and AD. According to the properties of midpoints, PS is parallel to side BD and half its length.
- Similarly, the segment QR joins the midpoints of sides BC and CD. According to the properties of midpoints, QR is parallel to side BD and half its length.
- Therefore, PS and QR are parallel and have equal lengths. This means that PQRS is a parallelogram.
4. Proving PQRS is a Rectangle (continued):
- Next, let's consider triangle ABD. Since R is the midpoint of CD, the segment QR joins the midpoints of sides CD and DA. According to the properties of midpoints, QR is parallel to side AB and half its length.
- Similarly, the segment PS joins the midpoints of sides AB and BC. According to the properties of midpoints, PS is parallel to side AB and half its length.
- Therefore, QR and PS are parallel and have equal lengths. This means that PQRS is a parallelogram.
5. Proving PQRS is a Rectangle (continued):
- From the previous steps, we have established that PQRS is a parallelogram.
- Now, let's consider the diagonals of PQRS. Diagonals PR and QS bisect each other at point O.
- Since PQRS is a parallelogram, its diagonals bisect each other.
- Moreover, since PR and QS are parallel, they are also perpendicular to each other.
- Therefore, PQRS is a parallelogram with perpendicular diagonals, which satisfies the definition of a rectangle.
Conclusion:
- We have proved that PQRS is a rectangle using the properties of a rhombus, the properties of midpoints, and the properties of a parallelogram with perpendicular diagonals.
- Thus, the claim is proven.
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ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle
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ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle 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 ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle.
ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle 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 ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a rhombus and P, Q, R and S are the mid point of resides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is rectangle.
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