Statement 1: The diagonals of a quadrilateral are perpendicular to each other; P, Q, R and S are the midpoints of sides AB, BC, CD and DA respectively.
Statement 2: Quadrilateral PQRS is a square.
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.

Question

Statement 1: The diagonals of a quadrilateral are perpendicular to each other; P, Q, R and S are the midpoints of sides AB, BC, CD and DA respectively.

The diagonals of a quadrilateral are perpendicular to each other; P, Q, R and S are the midpoints of sides AB, BC, CD and DA respectively. Quadrilateral PQRS is a square. Mid-Point Theorem, Concise Mathematics Solutions ICSE Class 9.

Statement 2: Quadrilateral PQRS is a square.

Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.

KnowledgeBoat · Accepted Answer

From figure, PA = PB ⇒ P is mid-point of AB SA = SD ⇒ S is mid-point of AD BQ = CQ ⇒ Q is mid-point of BC CR = RD ⇒ R is mid-point of CD So, statement 1 is true. By mid-point theorem, The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ ABC, P and Q are mid-points of AB and BC respectively. ⇒ PQ = AC and PQ || AC. [By mid-point theorem] ................(1) In △ ADC, S and R are mid-points of AD and CD respectively. ⇒ SR = AC and SR || AC. [By mid-point theorem] .................(2) From (1) and (2), we get : PQ = SR and PQ || SR. In △ BCD, R and Q are mid-points of CD and BC respectively. ⇒ QR = BD and QR || BD. [By mid-point theorem] .................(3) In △ ABD, S and P are mid-points of AD and AB respectively. ⇒ PS = BD and PS || BD. [By mid-point theorem] .................(4) From (3) and (4), we get : QR = PS and QR || PS. Since, diagonals of quadrilateral intersect at right angle. ∴ ∠AOD = ∠COD = ∠AOB = ∠BOC = 90°. From figure, PQ || AC ∴ ∠PXO = ∠AOD = 90° (Corresponding angles are equal) ∴ ∠QXO = ∠COD = 90° (Corresponding angles are equal) SR || AC ∴ ∠SZO = ∠AOB = 90° (Corresponding angles are equal) ∴ ∠RZO = ∠BOC = 90° (Corresponding angles are equal) PS || BD ∴ ∠S = ∠RZO = 90° (Corresponding angles are equal) ∴ ∠P = ∠QXO = 90° (Corresponding angles are equal) QR || BD ∴ ∠R = ∠SZO = 90° (Corresponding angles are equal) ∴ ∠Q = ∠PXO = 90° (Corresponding angles are equal) Since, in quadrilateral PQRS, Each interior angle is equal to 90° and opposite sides are parallel and equal, and we can prove that PQRS is a rectangle but cannot prove that all sides are equal. So, statement 2 is false. ∴ Statement 1 is true, and statement 2 is false. Hence, option 3 is the correct option.

Mathematics

Mid-point Theorem

Related Questions

In the given figure, AB || CD || EF and E is the mid-point of side AD, then :
OE : OF = 1 : 3
OE = OF
OF = 2 x OE
CF = FB

In rhombus PQRS; A, B and C are mid-points of sides PQ, QR and RS respectively. If ∠P = 60°, the angle PQR is equal to:
60°
90°
120°
none of these

Statement 1: AD is median of triangle ABC and DE is parallel to BA.
Statement 2: DE is median of triangle ADC.
Both the statements are true.
Both the statements are false.
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Mathematics

Mid-point Theorem

Related Questions

In the given figure, AB || CD || EF and E is the mid-point of side AD, then :OE : OF = 1 : 3OE = OFOF = 2 x OECF = FB

In rhombus PQRS; A, B and C are mid-points of sides PQ, QR and RS respectively. If ∠P = 60°, the angle PQR is equal to:60°90°120°none of these

Statement 1: AD is median of triangle ABC and DE is parallel to BA.Statement 2: DE is median of triangle ADC.Both the statements are true.Both the statements are false.Statement 1 is true, and statement 2 is false.Statement 1 is false, and statement 2 is true.

In the given figure, AB || CD || EF and E is the mid-point of side AD, then :
OE : OF = 1 : 3
OE = OF
OF = 2 x OE
CF = FB

In rhombus PQRS; A, B and C are mid-points of sides PQ, QR and RS respectively. If ∠P = 60°, the angle PQR is equal to:
60°
90°
120°
none of these

Statement 1: AD is median of triangle ABC and DE is parallel to BA.
Statement 2: DE is median of triangle ADC.
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.