Assertion (A): The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus.
Reason (R): Adjacent sides of a rhombus are equal and perpendicular to each other.
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false

Question

Assertion (A): The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus.

Reason (R): Adjacent sides of a rhombus are equal and perpendicular to each other.

The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus. R.S. Aggarwal Mathematics Solutions ICSE Class 9.

A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false

KnowledgeBoat · Accepted Answer

Join AC and BD. 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 midpoints of AB and BC respectively. ∴ PQ || AC and PQ = AC (By midpoint theorem) .....(1) Similarly in △ADC, S and R are midpoints of AD and CD respectively. ∴ RS || AC and RS = AC (By midpoint theorem) .....(2) In △ABD, P and S are midpoints of AB and AD respectively. ∴ PS || BD and PS = BD (By midpoint theorem) .....(3) Similarly in △BCD, Q and R are midpoints of BC and CD respectively. ∴ QR || BD and QR = BD (By midpoint theorem) .....(4) From (1) and (2) we get, PQ = RS and PQ || RS From (3) and (4) we get, PS = QR and PS || QR Since, opposite sides are parallel and equal. Thus, PQRS is a parallelogram. ∴ Assertion (A) is false. In rhombus, adjacent sides are equal and adjacent angles are supplementary. ∴ Reason (R) is false. Hence, option 4 is the correct option.

Mathematics

Mid-point Theorem

Related Questions

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order is a square only, if :
ABCD is a rhombus
Diagonals of ABCD are equal
Diagonals of ABCD are equal and perpendicular
Diagonals of ABCD are perpendicular

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF

Mathematics

Assertion (A): The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus.Reason (R): Adjacent sides of a rhombus are equal and perpendicular to each other.A is true, R is falseA is false, R is trueBoth A and R are trueBoth A and R are false

Mid-point Theorem

Related Questions

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order is a square only, if :ABCD is a rhombusDiagonals of ABCD are equalDiagonals of ABCD are equal and perpendicularDiagonals of ABCD are perpendicular

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :DE = EFAE = EF∠DAE = ∠EFC∠ADE = ∠ECF

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order is a square only, if :
ABCD is a rhombus
Diagonals of ABCD are equal
Diagonals of ABCD are equal and perpendicular
Diagonals of ABCD are perpendicular

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF