KnowledgeBoat Logo
|

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.

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.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Mid-point Theorem

4 Likes

Answer

Join AC and BD.

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.

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 = 12\dfrac{1}{2} AC (By midpoint theorem) …..(1)

Similarly in △ADC,

S and R are midpoints of AD and CD respectively.

∴ RS || AC and RS = 12\dfrac{1}{2} AC (By midpoint theorem) …..(2)

In △ABD,

P and S are midpoints of AB and AD respectively.

∴ PS || BD and PS = 12\dfrac{1}{2} BD (By midpoint theorem) …..(3)

Similarly in △BCD,

Q and R are midpoints of BC and CD respectively.

∴ QR || BD and QR = 12\dfrac{1}{2} 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.

Answered By

1 Like

Related Questions