**Quadrilaterals**

*NCERT Ex 8.2*
**Q1: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:**

(i) SR || AC and SR = AC

(ii)PQ = SR

(iii)PQRS is a parallelogram.
Answer:

(i) In ▲ADC, S and R are the mid-points of sides AD and CD respectively.

In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.

∴ SR || AC and SR =

AC ... (1)

(ii) In ▲ABC, P and Q are mid-points of sides AB and BC respectively. Therefore, by using mid-point theorem,

PQ || AC and PQ = AC ... (2)

Using equations (1) and (2), we obtain

PQ || SR and PQ = SR ... (3)

⇒ PQ = SR

(iii) From equation (3), we obtained

PQ || SR and PQ = SR

Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal.

Hence, PQRS is a parallelogram.

**Q2: ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.**
Answer:

In ▲ABC, P and Q are the mid-points of sides AB and BC respectively.

∴ PQ || AC and PQ =

AC (Using mid-point theorem) ... (1)