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 = $\dfrac{1}{2}$ AC and PQ || AC. [By mid-point theorem] …………….(1)

In △ ADC,

S and R are mid-points of AD and CD respectively.

⇒ SR = $\dfrac{1}{2}$ 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 = $\dfrac{1}{2}$ BD and QR || BD. [By mid-point theorem] ……………..(3)

In △ ABD,

S and P are mid-points of AD and AB respectively.

⇒ PS = $\dfrac{1}{2}$ 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

Answer

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.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.