In the trapezium ABCD, AB || DC and AB > DC. P and Q are the mid-points of the diagonals AC and BD. Then, PQ || AB and PQ = …..(AB - DC).

1. 2

2. 3

3. $\dfrac{1}{2}$

4. $\dfrac{1}{3}$

Case Study

Ankita took part in a Rangoli Competition. She made a beautiful Rangoli in the shape of a triangle ABC as shown. In the triangle, P, Q and R are mid-points of the sides AB, BC and CA respectively. She decorated it by putting a garland along the sides of △PQR. The lengths of the sides of the triangle are AB = 20 cm, BC = 26 cm and AC = 24 cm.

Based on this information, answer the following questions:

1. The length of AP is :
   (a) QB
   (b) QR
   (c) AR
   (d) RP

2. The length of PQ is :
   (a) 12 cm
   (b) 13 cm
   (c) 10 cm
   (d) 15 cm

3. The length of the garland is :
   (a) 30 cm
   (b) 32 cm
   (c) 35 cm
   (d) 40 cm

4. Area of △PQR is :
   (a) area of △ABC
   
   (b) $\dfrac{1}{2}$ area of △ABC
   
   (c) $\dfrac{1}{3}$ area of △ABC
   
   (d) $\dfrac{1}{4}$ area of △ABC
   
   

5. APQR is a :
   (a) Rectangle
   (b) Parallelogram
   (c) Square
   (d) Rhombus

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.

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

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order is a square only, if :

1. ABCD is a rhombus

2. Diagonals of ABCD are equal

3. Diagonals of ABCD are equal and perpendicular

4. Diagonals of ABCD are perpendicular